APPROXIMATING SYMMETRIC POSITIVE SEMIDEFINITE TENSORS OF EVEN ORDER

Abstract

Tensors of various orders can be used for modeling physical quantities such as strain and diffusion as well as curvature and other quantities of geometric origin. Depending on the physical properties of the modeled quantity, the estimated tensors are often required to satisfy the positivity constraint, which can be satisfied only with tensors of even order. Although the space ${\mathcal{P}}_0^{2m}$ of 2m^th-order symmetric positive semi-definite tensors is known to be a convex cone, enforcing positivity constraint directly on ${\mathcal{P}}_0^{2m}$ is usually not straightforward computationally because there is no known analytic description of ${\mathcal{P}}_0^{2m}$ for m > 1. In this paper, we propose a novel approach for enforcing the positivity constraint on even-order tensors by approximating the cone ${\mathcal{P}}_0^{2m}$ for the cases 0 < m < 3, and presenting an explicit characterization of the approximation Σ_2m ⊂ Ω_2m for m ≥ 1, using the subset ${\mathrm{\Omega }}_{2m}\subset {\mathcal{P}}_0^{2m}$ of semi-definite tensors that can be written as a sum of squares of tensors of order m. Furthermore, we show that this approximation leads to a non-negative linear least-squares (NNLS) optimization problem with the complexity that equals the number of generators in Σ_2m. Finally, we experimentally validate the proposed approach and we present an application for computing 2m^th-order diffusion tensors from Diffusion Weighted Magnetic Resonance Images.

1. Introduction

Multi-linear algebra is a generalization of linear algebra and tensors which are multi-linear forms are widely used for modeling various physical quantities commonly encountered in engineering and physics. Elasticity [34], stress, strain and diffusion [10] are some examples. In differential geometry, tensors are used to represent metrics, curvatures [40] and other geometric quantities. In image processing, structure tensors [46] have been used for texture analysis, trifocal tensors in multi-view geometry, etc. The tensors in most of these applications are required to satisfy certain properties. For example, the tensors that approximate the Bidirectional Reflectance Distribution Function (BRDF) [7] are anti-symmetric, while the diffusion [10] and the structure tensors [46] are antipodally symmetric. Furthermore, certain applications demand that the estimated tensors be positive-definite since they model positive-valued physical quantities such as the diffusivity function or the displacement probability of water molecules [8]. In this paper, we are interested in the case of fully symmetric positive-definite tensors of various orders and hence for sake of simplicity, every reference to the term tensor will imply this particular case of tensors unless otherwise stated.

Let denote the set of m^th-order symmetric positive-definite tensors in ℝ³. As is well-known, positivity condition requires the order m to be even. Denote ${\mathcal{P}}_0^m$ the closure of consisting of symmetric positive semi-definite tensors (PSD) in ℝ³. As subsets of the space of m^th-order symmetric tensors, , ${\mathcal{P}}_0^m$ are cones, convex subsets that are invariant under positive scaling [18]. In most applications, the main computational problem can be formulated as data interpolation problem with the domain being ${\mathcal{P}}_0^{2m}$. Specifically, the input data are often in the form {(x₁, y₁), ···, (x_k, y_k)} where x_i are directions in ℝ³ represented as points on the unit sphere S², and y_i are the values to be interpolated. The interpolation problem requires a non-negative tensor $\mathbf{T}\in {\mathcal{P}}_0^{2m}$ that interpolates the input data. Formulated as a least-squares problem, it has the form

$$\mathbf{T}=arg\underset{p\in {\mathcal{P}}_0^{2m}}{min}\sum _{i=1}^k{\mid y_i-p({\mathbf{x}}_i)\mid }^2.$$

We note that both the objective function and the domain ${\mathcal{P}}_0^{2m}$ are convex, and therefore, the optimization problem above is in fact a convex optimization problem that, in principle, can be solved using existing techniques [12]. However, a formal and significant difficulty of applying these methods is that except for the m = 1 case, there exists no known description of the cone ${\mathcal{P}}_0^{2m}$ as it is well-known that the positivity test for polynomials of degree m > 2 is a difficult problem. In the second-order case, the cone ${\mathcal{P}}_0^2$ is known to be self-dual in the sense that there exists an inner product < ·, · > on such that < A, B >≥ 0 for any $A,B\in {\mathcal{P}}_0^2$. The inner product allows the extension of the usual duality theory using Lagrange multipliers to the cone ${\mathcal{P}}_0^2$, and there is a well-developed theory of semi-definite programming (SDP) [12] that deals with linear objective functions on ${\mathcal{P}}_0^2$.

While the difficulty of providing a complete description of ${\mathcal{P}}_0^{2m}$ seems to be unsurmountable at this point, the main contribution of this paper is the realization of another formal difficulty that can be overcome relatively easily. A cone C in a vector space is said to be finitely-generated if there exists a finite number of elements v₁, ···, v_n ∈ C, its generators, such that every element c ∈ C can be written as a non-negative linear combination of the generators

$$c=a_1{v}_1+\cdots +a_n{v}_n,a_1,\cdots ,a_n\ge 0.$$

If the cone ${\mathcal{P}}_0^{2m}$ were finitely generated, the above optimization problem becomes a non-negative linear least-squares (NNLS) problem, with complexity (number of variables) equals to the number of generators. The advantage of solving an NNLS problem is that there are software packages that can efficiently solve NNLS problems containing thousands of variables [28]. While ${\mathcal{P}}_0^{2m}$ is not finitely-generated, it follows naturally that we can try to approximate ${\mathcal{P}}_0^{2m}$ with a finitely-generated subcone, and restrict the above optimization problem to the subcone. The restriction can be justified if the subcone can be shown to be a good approximation of ${\mathcal{P}}_0^{2m}$.

The second contribution of this paper is an explicit characterization of the approximations ${\mathrm{\sum }}_{2m}\subset {\mathcal{P}}_0^{2m}$ for 0 < m < 3, and Σ_2m ⊂ Ω_2m for m ≥ 1, where Σ_2m is a finitely-generated subcone in the respective spaces. More specifically, let Ω_2m denote the subcone in ${\mathcal{P}}_0^{2m}$ consisting of semi-definite tensors that can be written as a sum of squares of tensors of order m. We have the natural inclusions ${\mathrm{\sum }}_{2m}\subset {\mathrm{\Omega }}_{2m}\subset {\mathcal{P}}_0^{2m}$, and our result gives a detailed characterization of the approximation Σ_2m ⊂ Ω_2m in terms of the geometry of the generators in Σ_2m. In particular, for m = 1, 2, it is known that ${\mathrm{\Omega }}_{2m}={\mathcal{P}}_0^{2m}$, and our result then gives a detailed characterization of the approximation ${\mathrm{\sum }}_{2m}\subset {\mathcal{P}}_0^{2m}$. Our analysis have shown that, for the lower-order cases m = 1, 2, 3, which are of primary interest here, for a reasonable precision requirement, Ω_2m can be approximated by Σ_2m containing a few hundreds or at most a few thousands of generators. It follows that the corresponding NNLS problems have the complexity that are well within the capability of currently available NNLS algorithms [28]. We quantitatively validate our method via several experiments, and we also present an application of the proposed technique for estimating the diffusivity function from diffusion-weighted MRI to demonstrate both the efficiency and accuracy of the proposed method.

The rest of this paper is organized as follows: In Sec. 2, we define the finitely-generated subcone Σ_2m. We also develop the theory that quantifies the approximation Σ_2m ⊂ Ω_2m, and the main theorem proved in this section relates the approximation error with the geometry of the generators in Σ_2m. Using the theory developed in Sec. 2, in Sec. 3 we explicitly work out the formulas for the number of generators for Σ_2m required for a given accuracy requirement. The results show that, up to order-6 and depending on the order, it generally requires at most a few thousands of generators for Σ_2m in order to achieve a relative approximation error of less than 10%. Finally, in Sec. 4, we validate our theoretical findings using a set of experiments and we present an application of our method on diffusion-weighted MR datasets.

Related Work

Symmetric positive-definite (SPD) tensors of order-2 have been used in modeling the diffusivity function in the so called Diffusion Tensor MR Imaging (DT-MRI) [10]. SPD matrices can be endowed with a Riemannian metric that is invariant under affine transforms. This metric or its approximations have been employed for estimating and processing diffusion tensor fields [48, 47, 29, 38, 18, 9]. Tensors of 3^rd and 5^th order can model reflectance distributions with specularities and cast shadows in facial images and have been used for re-lighting in [7]. In general, odd-order tensors are generalizations of the order-1 tensor, which have been commonly used in computer graphics for representing the Lambertian reflectance model. Similarly, 4^th, 6^th or higher even-order tensors generalize the 2^nd-order tensors and have the ability to approximate multi-lobed functions [35, 30, 36] such as the kurtosis of diffusion [26]. In particular, some 4^th-order tensors can be expressed as 2^nd-order tensors in higher dimensions and their properties have been studied in detail by Moakher in [32, 33]. They however do not span the full space of the higher-order tensors as was shown in the case of order-4 tensors in [6, 5]. In [20], Ghosh et al. used the metric proposed by Moakher in [32, 33] to represent the space of 4^th-order SPD tensors using the geometry of 2^nd-order SPD tensors in higher dimensions. Recently, an algorithm for imposing positivity constraints on 4^th-order tensors using their equivalent ternary quartic polynomial representation was proposed in [6] and this was further developed in [5] and [21, 49].

After estimating a field of high-order tensors, it can be processed using a Finsler metric by appropriately modifying the polynomial equivalent representation of the tensors that satisfy the properties of Finsler geometry [4]. This method can be used for neuronal fiber tracking from high angular resolution diffusion MRI data. Further processing of higher-order tensor fields can be achieved by using the eigenvalue decomposition of matrices which has been extended for the case of high-order tensors in [23]. In this framework, the eigenvalues correspond to the extreme values (minima or maxima) of a tensor and they can be used to extract useful information from the kurtosis tensor [42] as well as the orientation of maximum diffusion [11, 22]. Another method for extracting the principal orientation of diffusion from a higher-order tensor was recently described in [44].

Although, high-order tensors have been employed in most of the aforementioned methods due to their simple polynomial form and their ability to model multi-lobed spherical functions, there are no existing methods for imposing positivity constraints in symmetric tensors of any order higher than two and four. The need to impose positivity constraints becomes essential especially in the case where the tensors approximate positive-valued physical quantities, and it has been shown that imposing the positivity constraint on the tensors approximating the diffusivity function being estimated reduces the approximation errors significantly [5]. Recently, Pasternak et al. [37] also emphasized the importance of enforcing positivity constraints in processing diffusion tensor MR images.

Finally, although Cartesian tensors basis have been widely used for modeling the diffusivity function in DW-MRI, we would like to mention that Spherical Harmonic basis have been employed in approximating other spherical functions involved in DW-MRI processing such as the diffusion propagator. A detailed review of several multi fiber reconstruction methods that employ spherical harmonic basis can be found in the recent article by Descoteaux et al. on Diffusion Propagator Imaging [17]. The orientation distribution function (ODF) is another example of a DW-MRI related spherical function, which can be reconstructed from Q-ball imaging data [16, 13, 3] and was recently done in [2] by using the mathematically correct definition of ODF and deriving a closed form expression for the same. In this article, however, our main focus is on the use of Cartesian tensor basis for parameterizing the diffusivity function in DW-MR datasets.

2. Theory

We will consider symmetric tensors of order m as functions defined on the unit sphere S² in ℝ³. In particular, symmetric tensors of order m can be identified with homogeneous polynomials of degree m: for a symmetric tensor T of order m, its associated homogeneous polynomial P(x, y, z) is given as

$$P(x,y,z)=\mathbf{T}(\underset{m}{\underbrace{\mathbf{x},\cdots ,\mathbf{x}}}),$$

where x = [x y z]^⊤. Under this identification, are homogeneous polynomials of degree m that do not vanish on S², and similarly, ${\mathcal{P}}_0^m$ are degree-m homogeneous polynomials that do not take negative values in ℝ³. Both are now considered as cones in H_m, the set of homogeneous polynomials of degree m. For even degree 2m, let Ω_2m denote the subset of ${\mathcal{P}}_0^{2m}$ consisting of polynomials that can be written as a sum of squares of polynomials of degree m. Ω_2m is clearly a subcone of ${\mathcal{P}}_0^{2m}$ for all m ≥ 1, and for m = 1, 2, it is known that ${\mathrm{\Omega }}_{2m}={\mathcal{P}}_0^{2m}$: the m = 1 case follows easily from linear algebra and m = 2 case is the content of Hilbert’s theorem on ternary quartics [24]. For m > 2, however, the inclusion is strict ${\mathrm{\Omega }}_{2m}⊊{\mathcal{P}}_0^{2m}$. In this section, we will describe a general method for approximating Ω_2m using a finitely-generated subcone Σ_2m in Ω_2m, and we will provide a characterization of the approximation error in terms of the geometry of the generators of Σ_2m. For the important quadratic and quartic cases m = 1, 2, our result provides an approximation of the full PSD cone ${\mathcal{P}}_0^{2m}$ using a finitely-generated subcone Ω_2m.

The basic norm used in this paper is the L¹-norm over the sphere S². More specifically, for any P ∈ H_m, its L¹-norm ||P ||₁ is the integral over S²

$${\left|\right|P\left|\right|}_1={\int }_{{\mathbf{S}}^2}\mid P(\mathbf{x})\mid d\mathbf{x}.$$

That it is indeed a norm follows from the fact that for two homogeneous polynomials P, Q, P = Q as polynomials if and only if ||P − Q||₁ = 0. Note that the other norm properties are trivial to prove. For any $P\in {\mathcal{P}}_0^{2m}$ and a subcone ${\mathrm{\sum }}_{2m}\subset {\mathcal{P}}_0^{2m}$, we define the relative L¹-approximation error of P as

$${\mathbf{E}}_{{\mathrm{\sum }}_{2m}}(P)=\frac{{min}_{p\in {\mathrm{\sum }}_{2m}}{\left|\right|P-p\left|\right|}_1}{{\left|\right|P\left|\right|}_1}.$$ (2.1)

Proposition 2.1

Let Σ_2m be a closed subcone in ${\mathcal{P}}_0^{2m}$ and $P\in {\mathcal{P}}_0^{2m}$.

1. The L¹-norm is convex: for any $p\in {\mathcal{P}}_0^{2m}$, the function g(q), $q\in {\mathcal{P}}_0^{2m}$

   $$g(q)={\left|\right|p-q\left|\right|}_1$$

   is a convex function on ${\mathcal{P}}_0^{2m}$.

2. For P ≠ 0, E_Σ2m(P) = 0 if and only if P ∈ Σ_2m. For any s > 0,

   $${\mathbf{E}}_{{\mathrm{\sum }}_{2m}}(sP)={\mathbf{E}}_{{\mathrm{\sum }}_{2m}}(P).$$

Proof

For any $q_1,q_2\in {\mathcal{P}}_0^{2m}$,

$$g({tq}_1+(1-t)q_2)={\int }_{{\mathbf{S}}^2}\mid p-{tq}_1-(1-t)q_2\mid d\mathbf{x}\le {\int }_{{\mathbf{S}}^2}\mid tp-{tq}_1\mid d\mathbf{x}+{\int }_{{\mathbf{S}}^2}\mid (1-t)p-(1-t)q_2\mid d\mathbf{x},$$

and the convexity of the norm on ${\mathcal{P}}_0^{2m}$ follows. (2) is clear because Σ_2m is closed. The invariance of E_Σ2m under positive scaling follows readily from the definition.

Let m₁(x), ···, m_d(m)(x) denote the $d(m)=\frac{(m+2)(m+1)}{2}$ monomials in H_m. (Note that d(m) also equals to the number of symmetric spherical harmonic basis elements, which can be mapped to the monomials in H_m using an one-to-one transformation [35, 15].) The monomials form a basis in H_m that identifies H_m with ℝ^d(m). We will denote HS_m the unit sphere in H_m, consisting of polynomials

$$p(\mathbf{x})=\sum _{i=1}^{d(m)}{a}_i{\mathbf{m}}_i(\mathbf{x})$$

such that $a_1^2+\cdots +a_{d(m)}^2=1$. The subcone Σ_2m will be defined using polynomials in HS_m, and this is accomplished through the square map ${\mathcal{F}}_m^2:{\mathbf{H}}_m\to {\mathbf{H}}_{2m}$:

$${\mathcal{F}}_m^2(p)=p^2.$$

Clearly ${\mathcal{F}}_m^2$ is a smooth map, and ${\mathcal{F}}_m^2(p)={\mathcal{F}}_m^2(q)$ if and only if p = ±q. While ${\mathcal{F}}_m^2$ is not linear, it maps rays in H_m to rays in ${\mathbf{H}}_{2m}:{\mathcal{F}}_m^2(tp)=t^2{\mathcal{F}}_m^2(p)$. The geometry of the map ${\mathcal{F}}_m^2$ will play a crucial role in our analysis below, and it is quantified by its condition number η_m. First, we define two quantities.

$${\eta }_m^{max}=\underset{p\in {\mathbf{HS}}_m}{max}{\left|\right|{\mathcal{F}}_m^2(p)\left|\right|}_1,{\eta }_m^{min}=\underset{p\in {\mathbf{HS}}_m}{min}{\left|\right|{\mathcal{F}}_m^2(p)\left|\right|}_1.$$

Clearly we have ${\eta }_m^{min}>0$ since HS_m does not contain the zero polynomial. The two numbers measure the amount of stretching and shrinking ${\mathcal{F}}_m^2$ does to the sphere HS_m. Their ratio gives the condition number η _m for ${\mathcal{F}}_m^2$

$${\eta }_m=\frac{{\eta }_m^{max}}{{\eta }_m^{min}}.$$

In the following, we will often drop the subscript and denote the condition number simply as η when the degree m in the context is clear. Figure 2.1 illustrates the effect of ${\mathcal{F}}_m^2$ and its condition number η.

Fig. 2.1.

Left: Comparison between ${\mathcal{F}}_1^2$ and ${\mathcal{F}}_2^2$. Let $S_r^{2m}$, r > 0 denote the circle with radius r centered at origin in H_2m. ${\mathcal{F}}_1^2$ is isotropic in the sense that ${\left|\right|{\mathcal{F}}_1^2(p)\left|\right|}_1\in S_{4\pi /3}^2$ for all p ∈ HS₁. ${\mathcal{F}}_2^2$, on the other hand, is not isotropic. HS₂ is the five-dimensional sphere S⁵. Its equator can be identified with S⁴, and the two polynomials $\pm 1/\sqrt{3}(x^2+y^2+z^2)$ form the two poles. Inside the equator, are embedded S¹ and S². ${\mathcal{F}}_2^2$ maps the poles ±(x² + y² + z²) to $S_{4\pi /3}^4$, and it maps the embedded S¹ and S² to $S_{8\pi /15}^4,S_{4\pi /15}^4$, respectively. The condition number η for ${\mathcal{F}}_2^2$ on HS₂ is 5. Restricting ${\mathcal{F}}_2^2$ to the equator S⁴, the condition number improves to 2. Right: Local and non-local approximations. For each p ∈ HS_m, Lemma 2.8 approximates p first with the vertices of the simplex containing p. This local approximation is further improved using non-local approximations as the polynomials (p_i − p_j)² are approximated by polynomials in HS_m that are generally far from p.

Proposition 2.2

${\eta }_m^{max},{\eta }_m^{min}$ and hence η can be determined by evaluating $\frac{d{(m)}^2+d(m)}{2}$ trigonometric integrals.

Proof

Let m₁, ···, m_d(m) denote the d(m) monomials in H_m. A polynomial p ∈ HS_m is identified with the vector of coefficients a = [a₁, ···, a_d(m)]^⊤ as p = a₁m_i + ··· + a_d(m)m_d(m). The L¹-norm || (p)||₁ is the integral of p² over S² that can be written as

$${\left|\right|{\mathcal{F}}_2(p)\left|\right|}_1=\sum _{i,j=1}^{d(m)}{a}_i{a}_j{\int }_{{\mathbf{S}}^2}{\mathbf{m}}_i(\mathbf{x}){\mathbf{m}}_j(\mathbf{x})d\mathbf{x}.$$

Let Λ^m denote the d(m) × d(m) matrix whose components ${\mathrm{\Lambda }}_{ij}^m$ are the integrals ∫_S² m_i(x)m_j(x) dx, we have

$${\left|\right|{\mathcal{F}}_2(p)\left|\right|}_1={\mathbf{a}}^{\top }{\mathrm{\Lambda }}^m\mathbf{a}.$$

It follows that ${\eta }_m^{max},{\eta }_m^{min}$ can be determined as

$${\eta }_m^{max}=\underset{{\mathbf{a}}^{\top }\mathbf{a}=1}{min}{\mathbf{a}}^{\top }{\mathrm{\Lambda }}^m\mathbf{a},{\eta }_m^{min}=\underset{{\mathbf{a}}^{\top }\mathbf{a}=1}{min}{\mathbf{a}}^{\top }{\mathrm{\Lambda }}^m\mathbf{a},$$

both of which can be solved once Λ^m is known using Singular Value Decomposition. The integrals ∫_S² m_i(x)m_j(x) dx can be computed in closed form since using spherical coordinates, x = sin ψ cos θ, y = sin ψ sin θ, z = cos ψ, each integral is a product of two trigonometric integrals

$${\int }_{{\mathbf{S}}^2}{\mathbf{m}}_i(\mathbf{x}){\mathbf{m}}_j(\mathbf{x})d\mathbf{x}=\left({\int }_{\theta =0}^{2\pi }{cos}^{b_1}\theta {sin}^{b_2}\theta d\theta \right)\left({\int }_{\psi =0}^{\pi }{cos}^{b_3}\psi {sin}^{b_4}\psi d\psi \right),$$

with exponents b₁, b₂, b₃, b₄ depending on m_i, m_j.

In practice, ${\mathrm{\Lambda }}_{ij}^m$ can be numerically evaluated to any desired accuracy without appealing to the closed-form integral formulas. Next we prove a simple result that partially explains why the linear case m = 1 is substantially easier than the nonlinear cases m > 1.

Proposition 2.3

η_m = 1 if and only if m = 1. That is, ${\mathcal{F}}_1^2$ is isotropic with respect to the L¹-norm in H₂.

Proof

The ‘if’ part follows readily from the fact that

$${\int }_{{\mathbf{S}}^2}xyd\mathbf{x}={\int }_{{\mathbf{S}}^2}xzd\mathbf{x}={\int }_{{\mathbf{S}}^2}yzd\mathbf{x}=0,$$

and

$${\int }_{{\mathbf{S}}^2}{x}^{2}d\mathbf{x}={\int }_{{\mathbf{S}}^2}{y}^{2}d\mathbf{x}={\int }_{{\mathbf{S}}^2}{z}^{2}d\mathbf{x}=\frac{4\pi }{3}.$$

The matrix Λ¹ is therefore diagonal with constant diagonal element $\frac{4\pi }{3}$, and ${\mathcal{F}}_1^2$ is isotropic with respect to the L¹-norm in H₂.

Conversely, for m > 1, let p = x^m, q = x^m−1y. We show that || (p)||₁ ≠ || (q)||₁:

$$\begin{array}{l}{\left|\right|{\mathcal{F}}_2(p)\left|\right|}_1={\int }_{{\mathbf{S}}^2}{x}^{2m}d\mathbf{x}={\int }_{\psi =0}^{\pi }{\int }_{\theta =0}^{2\pi }{sin}^{2m}\psi {cos}^{2m}\theta sin\psi d\theta d\psi ,\\ {\left|\right|{\mathcal{F}}_2(q)\left|\right|}_1={\int }_{{\mathbf{S}}^2}{x}^{2m-2}{y}^{2}d\mathbf{x}={\int }_{\psi =0}^{\pi }{\int }_{\theta =0}^{2\pi }{sin}^{2m}\psi {cos}^{2m-2}\theta {sin}^2\theta sin\psi d\theta d\psi .\end{array}$$

Let $c={\int }_{\psi =0}^{\pi }{sin}^{2m+1}\psi d\psi$, we have

$$\begin{array}{l}{\left|\right|{\mathcal{F}}_2(p)\left|\right|}_1=c{\int }_{\theta =0}^{2\pi }{cos}^{2\pi }\theta d\theta ,\\ {\left|\right|{\mathcal{F}}_2(q)\left|\right|}_1=c{\int }_{\theta =0}^{2\pi }{cos}^{2m-2}\theta {sin}^2\theta d\theta .\end{array}$$

Therefore,

$$\begin{array}{l}{\left|\right|{\mathcal{F}}_2(p)\left|\right|}_1-{\left|\right|{\mathcal{F}}_2(q)\left|\right|}_1=c{\int }_{\theta =0}^{2\pi }{cos}^{2m-2}\theta ({cos}^2\theta -{sin}^2\theta )d\theta \\ =c{\int }_{\theta =0}^{2\pi }{cos}^{2m-2}\theta (2{cos}^2\theta -1)d\theta .\end{array}$$

Since ${\int }_{\pi =0}^{2\pi }{cos}^n\theta d\theta =\frac{n-1}{n}{\int }_{\pi =0}^{2\pi }{cos}^{n-2}\theta d\theta$ for any n ≥ 2, we have

$${\left|\right|{\mathcal{F}}_2(p)\left|\right|}_1-{\left|\right|{\mathcal{F}}_2(q)\left|\right|}_1=c(\frac{2m-1}{m}-1){\int }_{\theta =0}^{2\pi }{cos}^{2m-2}\theta d\theta ,$$

which shows that || (p)||₁ − || (q)||₁ ≠ 0 if m > 1. This implies η_m > 1 if m > 1.

Using the square map ${\mathcal{F}}_m^2$, we will define the approximating subcone ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ by specifying its generators as polynomials in HS_m. More specifically, let = {p₁, ···, p_k} denote a finite set of k polynomials (points) in HS_m. Its associated cone ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ in H_2m is generated by the finite set of generators ${\mathcal{F}}_m^2(\mathcal{C})=\{p_1^2,\cdots ,p_k^2\}$: elements in ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ are non-negative linear combinations of ${\mathcal{F}}_m^2(p_i)$:

$$p=a_1{p}_1^2+\cdots +a_k{p}_k^2,$$

for some a₁, ···, a_k ≥ 0. It is immediately clear that ${\mathrm{\sum }}_{2m}^{\mathcal{C}}\subset {\mathrm{\Omega }}_{2m}\subset {\mathcal{P}}_0^{2m}$ for any finite subset ⊂ HS_m. Since ${\mathcal{F}}_m^2(p)={\mathcal{F}}_m^2(-p)$, we can restrict points in to lie in one chosen hemisphere of HS_m. For such , its completion ⊂ is obtained by joining all antipodal points of points in ,

$$\overline{\mathcal{C}}=\{p_1,\cdots ,p_k,-p_1,\cdots ,-p_k\}.$$

Examples

For m = 1, H₁ is ℝ³ and HS₁ is S². If consists of four points {[1, 0, 0]^⊤, [0, 1, 0]^⊤, [0, 0, 1]^⊤, [ $\sqrt{1/3},\sqrt{1/3},\sqrt{1/3}$]}, the four polynomials p₁, p₂, p₃, p₄ are x, y, z and $\sqrt{1/3}x+\sqrt{1/3}y+\sqrt{1/3}z$, respectively. Elements in ${\mathrm{\sum }}_2^{\mathcal{C}}$ are non-negative combinations of the four polynomials $p_1^2,p_2^2,p_3^2,p_4^2$. More precisely, any $p\in {\mathrm{\sum }}_2^{\mathcal{C}}$ is determined (in this case, uniquely) by four non-negative numbers a₁, a₂, a₃, a₄ ≥ 0 such that

$$p(x,y,z)=(a_1+\frac{a_4}{3})x^2+(a_2+\frac{a_4}{3})y^2+(a_3+\frac{a_4}{3})z^2+\frac{2a_4}{3}(xy+xz+yz).$$

For m = 2, H₂ can be identified with ℝ⁶ using the monomial basis {x², y², z², xy, xz, yz} and HS₂ is S⁵. If consists of three points

$${[\lambda ,\lambda ,0,0,0,0]}^{\top },{[\lambda ,0,0,-\lambda ,0,0]}^{\top },{[0,-\lambda ,0,0,0,\lambda ]}^{\top },$$

where $\lambda =\sqrt{1/2}$, the three polynomials p₁, p₂, p₃ are λ(x² + y²), λ(x² − xy), λ(yz − y²). Any $p\in {\mathrm{\sum }}_4^{\mathcal{C}}$ can be written (again uniquely) as

$$p(x,y,z)=\frac{(a_1+a_2)}{2}{x}^4+\frac{(a_1+a_3)}{2}{y}^4+(a_1+\frac{a_2}{2})x^2{y}^2-a_2{x}^{3}y-a_3{y}^{3}z+\frac{a_3}{2}{y}^2{z}^2$$

for three non-negative a₁, a₂, a₃.

The inclusion ${\mathrm{\sum }}_{2m}^{\mathcal{C}}\subset {\mathrm{\Omega }}_{2m}$ gives an approximation of Ω_2m by ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$, and it involves two main components: the square map ${\mathcal{F}}_m^2$ and the chosen polynomials in that provide the generators in ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ through ${\mathcal{F}}_m^2$. The main result of our analysis on the approximation error of ${\mathrm{\sum }}_{2m}^{\mathcal{C}}\subset {\mathrm{\Omega }}_{2m}$ is given in the next theorem, which asserts that the approximation error can be bounded by a product of contributions from both components: the condition number η_m of ${\mathcal{F}}_m^2$ and the condition number θ( ) of the set whose definition we now turn to.

Condition Number θ( ) of

We use θ( ) as the measure that quantifies the approximation of any q ∈ HS_m, considered as a point on the sphere, by the finite set . We will use the spherical distance d_HSm(p, q) (arc-length in radians) to measure the distance between a pair of points p, q on the sphere HS_m, and in particular, d_HSm(p, q) is the angle between the two unit vectors p, q in HS_m. A set is said to be good if there is a triangulation of HS_m as a simplicial complex whose vertex set is the completion of . Since HS_m has dimension d(m) − 1, the top-dimensional simplexes in have dimension d(m) − 1 as well. Therefore, for any q ∈ HS_m, there is a d(m) − 1-simplex σ ∈ containing q. In particular, we will assume that q can be written as a non-negative linear combination of the vertices of σ: q = a₀p₀ + ··· + a_d(m)−1p_d(m)−1 with a₀, ···, a_d(m)−1 ≥ 0. While this is in general not true for an arbitrary triangulation of HS_m, it is not difficult to show that can be modified (without changing its underlying abstract simplicial complex) to satisfy this property, e.g., by first defining a triangulation of the vertices in considered as points in the Euclidean space ℝ^d(m) using the same abstract simplicial complex as and radially projecting the simplices onto HS_m. For 0 ≤ k ≤ d(m) − 1, will denote the set of k-simplices in , and for a k-simplex σ ∈ , its width δ(σ) is defined as the maximal distance between its vertices, p₀, · · ·, p_k,

$$\delta (\sigma )=\underset{0\le i,j\le k}{max}{\mathbf{d}}_{{\mathbf{HS}}_m}(p_i,p_j).$$

For a triangulation , we define its width to be the maximal width of its top-dimensional simplices:

$$\sigma (\mathcal{T})=\underset{\sigma \in {\mathcal{T}}^{d(m)-1}}{max}\delta (\sigma ).$$

The condition number of is then defined as the minimal width of the triangulations that have as its vertex set:

$$\sigma (\mathcal{C})=\underset{\mathcal{T},{\mathcal{T}}^0=\overline{\mathcal{C}}}{min}\delta (\mathcal{T}).$$

Since is finite, there exists a triangulation Δ( ) whose width gives the condition number θ( ). We note that 0 < θ( ) < π, and for a good set , the following conditions hold,

1. For each q ∈ HS_m, there are d(m) elements, p₀, ···, p_d(m)−1, in such that q = a₀p₀ + ··· a_d(m)−1p_d(m)−1 for a₀, ···, a_d(m)−1 ≥ 0 and d_HSm(p_i, p_j) < θ( ) for any 0 ≤ i, j ≤ d(m).

2. For each q ∈ HS_m, there exists p ∈ such that d_HSm(q, p) < θ( ).

Property (1) follows immediately from the definition. Property (2) can be shown to follow from the requirement that if q ∈ σ ∈ , q is a non-negative linear combination of vertices in σ.

Theorem 2.4

Let denote a good finite subset in HS_m and ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ its associated finitely-generated subcone in H_2m. Let θ = θ( ) denote the condition number of as defined above and η_m the condition number of ${\mathcal{F}}_m^2$. Then, for any polynomial r ∈ Ω_2m, its L¹-relative approximation error ${\mathbf{E}}_{{\mathrm{\sum }}_{2m}^{\mathcal{C}}}(r)$ satisfies

$${\mathbf{E}}_{{\mathrm{\sum }}_{\mathcal{C}}^{2m}}(r)\le 4tan\theta {sin}^2\frac{\theta }{2}{\eta }_m^2.$$

The bound above constitutes our quantitative characterization of the approximation ${\mathrm{\sum }}_{2m}^{\mathcal{C}}\subset {\mathrm{\Omega }}_{2m}$. Not surprisingly, the bound provided above depends on both the map ${\mathcal{F}}_m^2$ as well as the set through θ and η. The error measured by ${\mathbf{E}}_{{\mathrm{\sum }}_{\mathcal{C}}^{2m}}$ takes place in H_2m, and the bound on the right factored into two components with contribution from θ that essentially measures how well an arbitrary point q ∈ HS_m can be approximated using and its associated triangulation Δ( ). In particular, as will be seen from the proof, tan θ arises from approximating q using its nearest neighbor in as in Property (2) above while ${sin}^2\frac{\theta }{2}$ comes from approximating q using the simplex σ containing it as in Property (1).

We will prove the theorem through a sequence of lemmas given below. However, before delving into the proof, we remark that although using the triangulation Δ( ) to define θ( ) may seem unnecessary at first, it is in fact crucial to have Property (1) in order to produce a smaller bound on the error. For example, it is possible to define θ( ) using only Property (2), i.e., each q ∈ HS_m can be approximated by a p ∈ such that d_HSm(q, p) < θ( ). However, this hypothesis itself is only strong enough to produce the bound given in Lemma 2.6 (Equation 2.4). Disregarding η_m, the bound given in Equation 2.4 is 2 sin θ, which is considerably inferior to the bound of 4 tan θ sin² given in Theorem 2.4. In particular, for small θ, the former is approximately 2θ while the latter is θ³ (See Equation 3.1), two order of magnitude less. As will be clear in the proof, the main issue is to approximate the polynomial q² for any q ∈ HS_m with a sum of squares of polynomials in . Using only Property (2), it is difficult to determine what polynomials in HS_m can be used to approximate q² other than the polynomial p ∈ that is closest to q. With Property (1), we have more choices at our disposal as we can approximate q² using the vertices p_i of the simplex σ that contains q, and more importantly, the remainder of this approximation (sum of (p_i − p_j)²) can be further approximated using polynomials in . This is the content of Lemma 2.8. In particular, when approximating q², Property (1) allows the access of not only the polynomials p_i ∈ that are neighbors of q but also polynomials in that are usually far away from q. See Figure 2.1. Furthermore, as will be detailed in Section 3, Property (1) allows us to formulate a simple method for estimating the minimal number of points (polynomials) in needed for a given precision requirement.

Lemma 2.5

Let p, q be two polynomials in HS_m and θ = d_HSm(p, q) denote their geodesic distance considered as points on the sphere HS_m. We have

$${\int }_{{\mathbf{S}}^2}{\mid p(\mathbf{x})-q(\mathbf{x})\mid }^{2}d\mathbf{x}\le 4{sin}^2\frac{\theta }{2}{\eta }_m^{max}.$$

Proof

Let r = p − q. As a vector in H_m, |r| = |p − q|. Using the law of cosines,

$$\gamma =\mid r\mid =\mid p-q\mid =\sqrt{2-2cos\theta }=2sin\frac{\theta }{2}.$$ (2.2)

Therefore, r/γ ∈ HS_m, and we have

$${\int }_{{\mathbf{S}}^2}{r}^2(\mathbf{x})d\mathbf{x}={\gamma }^2{\int }_{{\mathbf{S}}^2}{(r(\mathbf{x})/\gamma )}^{2}d\mathbf{x}\le {\gamma }^2{\eta }_m^{max},$$

and the result follows.

Next we prove an important lemma which shows that for two nearby p, q in HS_m, we can approximate q² using p² such that the L¹-approximation error is a fraction (depending on the geodesic distance) of the L¹-norm of p².

Lemma 2.6

Let p, q be two polynomials in HS_m and θ = d_HSm(p, q) denote their geodesic distance considered as points on the sphere HS_m. Let || (p) − (q)||₁ denote the L¹-difference between (p), (q)

$${\left|\right|{\mathcal{F}}_2(p)-{\mathcal{F}}_2(q)\left|\right|}_1={\int }_{{\mathbf{S}}^2}\mid p^2(\mathbf{x})-q^2(\mathbf{x})\mid d\mathbf{x}.$$ (2.3)

We have

$${\left|\right|{\mathcal{F}}_2(p)-{\mathcal{F}}_2(q)\left|\right|}_1\le 2sin\theta {\eta }_m{\left|\right|{\mathcal{F}}_2(p)\left|\right|}_1.$$ (2.4)

Proof

Using Hölder’s inequality, we have

$$\begin{array}{l}{\int }_{{\mathbf{S}}^2}\mid p^2(\mathbf{x})-q^2(\mathbf{x})\mid d\mathbf{x}={\int }_{{\mathbf{S}}^2}\mid p(\mathbf{x})-q(\mathbf{x})\mid \mid p(\mathbf{x})+q(\mathbf{x})\mid d\mathbf{x}\\ \le {\left({\int }_{{\mathbf{S}}^2}{\mid p(\mathbf{x})-q(\mathbf{x})\mid }^{2}d\mathbf{x}\right)}^{\frac{1}{2}}{\left({\int }_{{\mathbf{S}}^2}{\mid p(\mathbf{x})+q(\mathbf{x})\mid }^{2}d\mathbf{x}\right)}^{\frac{1}{2}}.\end{array}$$

The proof will proceed to bound the two terms on the right. By the preceding lemma, we have

$${\left({\int }_{{\mathbf{S}}^2}{\mid p(\mathbf{x})-q(\mathbf{x})\mid }^{2}d\mathbf{x}\right)}^{\frac{1}{2}}\le 2sin\frac{\theta }{2}\sqrt{{\eta }_m^{max}}.$$

For the second term, we will consider the polynomial $r=\gamma \frac{p+q}{2}$, where γ > 1 ensures that r ∈ HS_m. A quick calculation shows that $\gamma =1/cos\frac{\theta }{2}$. Since, by definition,

$${\int }_{{\mathbf{S}}^2}{r}^2(\mathbf{x})d\mathbf{x}\le {\eta }_m^{max},$$

we have

$${\left({\int }_{{\mathbf{S}}^2}{\mid p(\mathbf{x})+q(\mathbf{x})\mid }^{2}d\mathbf{x}\right)}^{\frac{1}{2}}={\left({\int }_{{\mathbf{S}}^2}\frac{4}{{\gamma }^2}{r}^2(\mathbf{x})d\mathbf{x}\right)}^{\frac{1}{2}}\le \frac{2}{\gamma }\sqrt{{\eta }_m^{max}}.$$

Combining the two inequalities, we have

$${\int }_{{\mathbf{S}}^2}\mid p^2(\mathbf{x})-q^2(\mathbf{x})\mid d\mathbf{x}\le 4cos\frac{\theta }{2}sin\frac{\theta }{2}{\eta }_m^{max}=2sin\theta {\eta }_m^{max}.$$

Since

$${\left|\right|{\mathcal{F}}_2(p)\left|\right|}_1={\int }_{{\mathbf{S}}^2}{p}^2(\mathbf{x})d\mathbf{x}\le {\eta }_m^{min},$$

it follows that

$${\left|\right|{\mathcal{F}}_2(p)-{\mathcal{F}}_2(q)\left|\right|}_1\le 2sin\theta \frac{{\eta }_m^{max}}{{\eta }_m^{min}}{\left|\right|{\mathcal{F}}_2(p)\left|\right|}_1.$$

This completes the proof.

We will use the preceding lemma to prove two basic error estimates. For any two points p, q in , the following lemma provides a bound on the approximation error for points that lie on the arc (geodesic path) joining p, q.

Lemma 2.7

Let p, q be two neighboring points in Δ( ), i.e., there is a 1-simplex σ¹ in Δ( ) with p, q as its two vertices. Let r = ap + bq be a convex combination of p, q with a, b ≥ 0 and a + b = 1. If θ = θ( ) denotes the condition number of , then

$${\mathbf{E}}_{{\mathrm{\sum }}_{2m}^{\mathcal{C}}}(r^2)\le 2sin\theta {tan}^2\frac{\theta }{2}{\eta }_m^2.$$

Proof

By definition of θ, d_HSm(p, q) ≤ θ. Let ϕ = a²p² + b²q² + abp² + abq² be an element in ${\mathrm{\sum }}_{\mathcal{C}}^{2m}$. We have

$$\begin{array}{l}\varphi -r^2=(a^2{p}^2+b^2{q}^2+{\mathit{abp}}^2+{\mathit{abq}}^2)-{(ap+bq)}^2\\ =ab{(p-q)}^2.\end{array}$$

Let γ = |p − q| and t = (p − q)/γ ∈ HS_m. There exists s ∈ such that the geodesic distance between t and s is less than θ. By the preceding lemma,

$${\int }_{{\mathbf{S}}^2}\mid t^2(\mathbf{x})-s^2(\mathbf{x})\mid d\mathbf{x}\le 2sin\theta {\eta }_m{\left|\right|t^2(\mathbf{x})\left|\right|}_1.$$

Now let φ = ϕ + abγ²s² be another element in ${\mathrm{\sum }}_{\mathcal{C}}^{2m}$. We have

$${\int }_{{\mathbf{S}}^2}\mid r^2(\mathbf{x})-\phi (\mathbf{x})\mid d\mathbf{x}=ab{\int }_{{\mathbf{S}}^2}\mid {(\gamma t)}^2-{(\gamma s)}^2\mid d\mathbf{x}\le 2\mathit{ab}sin\theta {\eta }_m{\left|\right|{(\gamma t)}^2(\mathbf{x})\left|\right|}_1.$$

By Lemma 2.5, ${\mid {(\gamma t)}^2(\mathbf{x})\mid }_1\le 4{sin}^2\frac{\theta }{2}{\eta }_m^{max}$. This gives

$${\int }_{{\mathbf{S}}^2}\mid r^2(\mathbf{x})-\phi (\mathbf{x})\mid d\mathbf{x}\le 2sin\theta {sin}^2\frac{\theta }{2}{\eta }_m{\eta }_m^{max}.$$ (2.5)

as $ab\le \frac{1}{4}$ for a, b ≥ 0 and a + b = 1. We next bound the L¹-norm of r²(x). Since r = ap + bq, there exists $1\le \gamma \le 1/cos\frac{\theta }{2}$ such that γr ∈ HS_m. This implies that

$${\int }_{WS}{\gamma }^2{r}^2(\mathbf{x})d\mathbf{x}\ge {\eta }_m^{min},$$

or

$${\int }_{{\mathbf{S}}^2}{r}^2(\mathbf{x})d\mathbf{x}\ge {cos}^2\frac{\theta }{2}{\eta }_m^{min}.$$ (2.6)

Combining Equations 2.5 and 2.6 gives the desired result.

The preceding lemma can be generalized immediately to higher-order convex combinations.

Lemma 2.8

Let p₁, ···, p_k denote the vertices of a k − 1-simplex σ^k−1 in Δ( ) as well as the corresponding homogeneous polynomials in HS_m. Let r = a₁p₁ + ··· +a_kp_k be a convex combination of p₁, ···, p_k with a₁, ···, a_k ≥ 0 and a₁ + ··· + a_k = 1. If θ denote the condition number of , Then

$${\mathbf{E}}_{{\mathrm{\sum }}_{2m}^{\mathcal{C}}}(r^2)\le 4tan\theta {sin}^2\frac{\theta }{2}{\eta }_m^2.$$

Proof

Expanding r², we have

$$r^2=\sum _{i=1}^k{a}_i^2{p}_i^2+2\sum _{i<j}{a}_i{a}_j{p}_i{p}_j.$$

The second sum contains $C_2^k=\frac{k(k-1)}{2}$ terms. To approximate r² using an element $\phi \in {\mathrm{\sum }}_{\mathcal{C}}^{2m}$, we proceed similarly as before. We start with ϕ equals the first sum above. For each cross-term 2a_ia_jp_ip_j in the second sum, we add $a_i{a}_j(p_i^2+p_j^2)$ to ϕ. This gives

$$\varphi =\sum _{i=1}^k{a}_i^2{p}_i^2+\sum _{i\le j}{a}_i{a}_j(p_i^2+p_j^2).$$

It follows that

$$\varphi -r^2=\sum _{i<j}{a}_i{a}_j{(p_i+p_j)}^2.$$

Next, we will approximate the squares (p_i − p_j)² using elements in ${\mathrm{\sum }}_{\mathcal{C}}^{2m}$ exactly as before. More specifically, let γ_ij = |p_i − q_j| and t_ij = (p_i − q_j)/γ_ij. There exists s_ij ∈ such that the geodesic distance between t_ij and s_ij is less than θ. Now let φ = ϕ + Σ_i<j a_ib_j(γ_ijs_ij)² be an element in ${\mathrm{\sum }}_{\mathcal{C}}^{2m}$. We have

$${\int }_{{\mathbf{S}}^2}\mid r^2(\mathbf{x})-\phi (\mathbf{x})\mid d\mathbf{x}\le \sum _{i<j}{a}_i{a}_j{\int }_{{\mathbf{S}}^2}\mid {({\gamma }_{ij}{t}_{ij})}^2(\mathbf{x})-{({\gamma }_{ij}{s}_{ij})}^2(\mathbf{x})\mid d\mathbf{x}.$$

It follows from Equations 2.2 and 2.4 that all the integrals on the right can be uniformly bounded

$${\int }_{{\mathbf{S}}^2}\mid {({\gamma }_{ij}{t}_{ij})}^2(\mathbf{x})-{({\gamma }_{ij}{s}_{ij})}^2(\mathbf{x})\mid d\mathbf{x}\le 8sin\theta {sin}^2\frac{\theta }{2}{\eta }_m^{max}{\eta }_m,$$

and this gives

$${\int }_{{\mathbf{S}}^2}\mid r^2(\mathbf{x})-\phi (\mathbf{x})\mid d\mathbf{x}\le 8sin\theta {sin}^2\frac{\theta }{2}{\eta }_m^{max}{\eta }_m\sum _{i<j}{a}_i{a}_j.$$

Since a₁ + ··· + a_k = 1,

$$\begin{array}{l}\sum _{i\le j}{a}_i{a}_j=\frac{{(a_1+\cdots +a_k)}^2-(a_1^2+\cdots +a_k^2)}{2}=\frac{1-(a_1^2+\cdots +a_k^2)}{2}\\ \le \frac{1-\frac{1}{k}}{2}=\frac{k-1}{2k}<\frac{1}{2}\end{array}$$ (2.7)

as $a_1^2+\cdots +a_k^2\ge \frac{1}{k}$ by Cauchy-Schwarz inequality. This yields the bound

$${\int }_{{\mathbf{S}}^2}\mid r^2(\mathbf{x})-\phi (\mathbf{x})\mid d\mathbf{x}\le 4sin\theta {sin}^2\frac{\theta }{2}{\eta }_m^{max}{\eta }_m.$$ (2.8)

We next bound the L¹-norm of r². Given that r = a₁p₁ + ··· a_kp_k, the following lemma shows that the L²-magnitude |r| of the vector r satisfies

$$\mid r\mid \ge \sqrt{cos\theta }.$$

Hence, there exists $1\le \gamma \le \frac{1}{\sqrt{cos\theta }}$ such that γr ∈ HS_m. Exactly as before, we have

$${\int }_{{\mathbf{S}}^2}{r}^2(\mathbf{x})d\mathbf{x}\ge \frac{1}{{\gamma }^2}{\eta }_m^{min}\ge cos\theta {\eta }_m^{min}.$$ (2.9)

Equations 2.8 and 2.9 together complete the proof.

Lemma 2.9

Let Δ denote a k-simplex in ℝ^d(m) whose vertices p₀, ···, p_k are on the unit sphere, i.e., ||p₀||₂ = ··· = ||p_k||₂ = 1. If there exists some α such that 1 > α > 0 and $p_i^{\top }{p}_j\ge \alpha$ for all i ≠ j, then for any x ∈ Δ,

$${\left|\right|\mathbf{x}\left|\right|}_2>\sqrt{\alpha }.$$

Proof

Let x = a₁p₁ + ··· a_kp_k with a_i ≥ 0 and a₁ + ··· + a_k = 1. It follows that

$${\mathbf{x}}^{\top }\mathbf{x}\ge \sum _{i=0}^k{a}_i^2+2\alpha \sum _{i<j}{a}_i{a}_j.$$

Let s = 2Σ_i<j a_ia_j and the above inequality becomes x^⊤x ≥ 1 − (1 − α)s. From Equation 2.7, we have $0\le s\le \frac{k-1}{k}<1$. It follows that

$${\mathbf{x}}^{\top }\mathbf{x}>1-(1-\alpha )=\alpha .$$

We remark that when k = 2, $\frac{k-1}{k}=\frac{1}{2}$ and the bound becomes tighter ${\mathbf{x}}^{\top }\mathbf{x}\ge \frac{1}{2}+\frac{\alpha }{2}$. This gives the cos θ term in Equation 2.9. Finally, we are ready to complete the proof of Theorem 2.4:

Proof

Since r(x) can be written as a sum of squares, by Proposition 2.10, it can be written as a sum of no more than d(m) terms with p_i ∈ HS_m:

$$r(\mathbf{x})=\sum _{i=1}^{d(m)}{a}_i{p}_i^2(\mathbf{x}).$$

Each p_i belongs to a (d(m) − 1)-dimensional simplex σ_i ∈ Δ( ). By the preceding lemma, each $p_i^2$ can be approximated by an element p̃i in ${\mathrm{\sum }}_{\mathcal{C}}^{2m}$ with uniformly bounded relative L¹-error

$${\left|\right|p_i^2(\mathbf{x})-{\stackrel{\sim }{p}}_i(\mathbf{x})\left|\right|}_1\le C{\left|\right|p_i^2(\mathbf{x})\left|\right|}_1,$$

where $C=4tan\theta {sin}^2\frac{\theta }{2}{\eta }_m^2$. Define $\stackrel{\sim }{r}\in {\mathrm{\sum }}_{\mathcal{C}}^{2m}$ as

$$\stackrel{\sim }{r}=\sum _{i=1}^{d(m)}{a}_i{\stackrel{\sim }{p}}_i,$$

and we have

$${\left|\right|r(\mathbf{x})-\stackrel{\sim }{r}(\mathbf{x})\left|\right|}_1\le \sum _{i=1}^{d(m)}{a}_i{\left|\right|p_i^2(\mathbf{x})-{\stackrel{\sim }{p}}_i(\mathbf{x})\left|\right|}_1\le C\sum _{i=1}^{d(m)}{a}_i{\left|\right|p_i^2(\mathbf{x})\left|\right|}_1.$$

On the other hand, we also have

$${\left|\right|r(\mathbf{x})\left|\right|}_1\le \sum _{i=1}^{d(m)}{a}_i{\int }_{{\mathbf{S}}^2}{p}_i^2(\mathbf{x})d\mathbf{x}=\sum _{i=1}^{d(m)}{a}_i{\left|\right|p_i^2(\mathbf{x})\left|\right|}_1$$

Combining both inequalities yields the desired result.

In the proof above we made use of the following proposition.

Proposition 2.10

Let r denote a homogeneous polynomial of degree 2m that can be written as a sum of squares of homogeneous polynomials of degree m. Then, r can be written as a sum of at most d(m) squares

$$r(\mathbf{x})=\sum _{i=1}^{d(m)}{a}_i{p}_i^2(\mathbf{x}),$$

where a₁, ···, a_d(m) ≥ 0 and p₁, ···, p_d(m) ∈ HS_m.

Proof

Suppose r is a sum of k squares of homogeneous polynomials q̃₁, ···, q̃_k of degree m

$$r(\mathbf{x})={\stackrel{\sim }{q}}_1^2(\mathbf{x})+\cdots +{\stackrel{\sim }{q}}_k^2(\mathbf{x}).$$

Denote m₁, ···, m_d(m) the d(m) monomials of degree m, and X the vector

$$\mathbf{X}={[{\mathbf{m}}_1(\mathbf{x}),{\mathbf{m}}_2(\mathbf{x}),\cdots ,{\mathbf{m}}_{d(m)}(\mathbf{x})]}^{\top }$$

whose components are the monomials. It follows that ${\stackrel{\sim }{q}}_i(\mathbf{x})={\mathbf{a}}_i^{\top }\mathbf{X}$ with a_i the vector whose components are coefficients of q̃_i(x), and

$$r(\mathbf{x})={\mathbf{X}}^{\top }({\mathbf{a}}_1{\mathbf{a}}_1^{\top }+\cdots +{\mathbf{a}}_k{\mathbf{a}}_k^{\top })\mathbf{X}={\mathbf{X}}^{\top }\mathbf{SX}.$$

The matrix S is symmetric and positive semi-definite with non-negative eigenvalues. Let λ₁, ··· λ_d(m) denote its complete set of eigenvalues and v₁, ···, v_d(m) their associated unit eigenvectors, |v_i|₂ = 1. It follows that

$$\mathbf{S}={\lambda }_1{\mathbf{v}}_1{\mathbf{v}}_1^{\top }+\cdots +{\lambda }_{d(m)}{\mathbf{v}}_{d(m)}{\mathbf{v}}_{d(m)}^{\top },$$

and

$$\begin{array}{l}r(\mathbf{x})={\lambda }_1{\mathbf{X}}^{\top }{\mathbf{v}}_1{\mathbf{v}}_1^{\top }\mathbf{X}+\cdots +{\lambda }_{d(m)}{\mathbf{X}}^{\top }{\mathbf{v}}_{d(m)}{\mathbf{v}}_{d(m)}^{\top }\mathbf{X}\\ ={\lambda }_1{q}_1^2(\mathbf{x})+\cdots {\lambda }_{d(m)}{q}_{d(m)}^2(\mathbf{x}),\end{array}$$

where $q_i(\mathbf{x})={\mathbf{v}}_i^{\top }\mathbf{X}\in {\mathbf{HS}}_m$ as |v_i|₂ = 1 for i = 1, ···, d(m).

3. Approximating PSD Tensors of Orders two, four and six

In this section, we apply Theorem 2.4 to derive formulas for the minimal number of generators in ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ needed to ensure that the approximation ${\mathrm{\sum }}_{2m}^{\mathcal{C}}\subset {\mathrm{\Omega }}_{2m}$ is within a given accuracy requirement. Specifically, the accuracy requirement is presented in the form of the relative L¹-approximation error ${\mathbf{E}}_{2m}^{\mathcal{C}}$ (cf. Equation 2.1): for 0 < ε < 1, we derive a formula that gives the (approximated) minimal number (ε, m) of generators in ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ such that any r ∈ Ω_2m can be approximated within ε using ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$, i.e.,

$${\mathbf{E}}_{2m}^{\mathcal{C}}(r)<\epsilon .$$

For PSD ternary tensors of orders two and four, it is known that they can be written as sums of squares of three tensors of order one and two, respectively. This follows from the well-known result that any ternary positive semi-definite homogeneous polynomial p(x) of degree two and four can be written as a sum of three squares of polynomials of degree one and two, respectively. The quadratic case follows easily from linear algebra while the quartic case follows from the celebrated theorem of Hilbert on ternary quartics [24]. We will first describe a general method for obtaining the formula (ε, m) for any order m, and we will then explicitly work out the three cases m = 1, 2, 3 that are of most interest for various applications.

3.1. Preliminaries

Given a required precision ε > 0, the bound provided by Theorem 2.4 allows us to determine the condition number θ = θ( ) for the point set in HS_m to ensure that the precision requirement is satisfied. The main result in this section is a simple estimate on the number (ε, m) of points in needed to achieve the desired θ on the sphere HS_m. Let $C_{\eta }(\theta )=4tan\theta {sin}^2\frac{\theta }{2}{\eta }^2$ denote the bound given in Theorem 2.4. Since

$$tan\theta {sin}^2\frac{\theta }{2}=\frac{1}{2}(tan\theta -sin\theta ),$$

C_η(θ) is a monotonically increasing function for $0\le \theta \le \frac{\pi }{2}$, and we will denote its inverse by $f_{\eta }(\epsilon )=C_{\eta }^{-1}(\epsilon )$. f_η can be numerically evaluated and the plots for f_η over the range 0.01 ≤ ε ≤ 0.1 for several different η-values are shown in Figure 3.1. If θ is assumed to be small,

Fig. 3.1.

Left: Plots of f_η for η = 1, 2, 4 in red, blue and green, respectively. ε varies from 0.01 to 0.1 and θ is given in degree. Right: Comparison plot of (ε, 1) according to Equations 3.7 (in red) and 3.8 (in blue). The estimate using Equation 3.7 is between 17% and 20% less than the estimate using Equation 3.8.

$$tan\theta {sin}^2\frac{\theta }{2}\approx \frac{{\theta }^3}{4}.$$ (3.1)

Therefore, $4tan\theta {sin}^2\frac{\theta }{2}{\eta }^2\approx \epsilon$ implies that

$$\theta \approx {(\frac{\epsilon }{{\eta }^2})}^{\frac{1}{3}}.$$ (3.2)

The formula above gives an estimate on the condition number θ = θ( ) given ε and η. We next give an estimate on the size of for the given θ( ). Let n = d(m) − 1 denote the dimension of the sphere HS_m and Δ( ) denote the triangulation associated with . A simplex in Δ( ) is said to be θ-regular if the distance between any pair of its vertices equals θ, and the edge joining any pair of vertices is a geodesics on HS_m. Due to the curvature on the sphere HS_m, it is not possible to cover HS_m with only θ-regular simplices. Therefore, we assume that the n-simplices in Δ( ) are approximately θ-regular in the sense that the geodesic distance between any pair of vertices of a n-simplex in HS_m is approximately θ and the edge joining them is approximately a geodesic as well. For each vertex v in Δ( ), its degree is the number of n-dimensional simplices having it as a vertex. To estimate the number of points in , we will estimate two quantities: the number K of n-dimensional simplices in Δ( ) and the average degree ν of the vertices. The number of points in can then be estimated as

$$\mathcal{N}(\epsilon ,m)=\#\text{of}\text{points}\text{in}\mathcal{C}\simeq \frac{(n+1)K}{2\nu }.$$

The occurrence of 2 in the denominator accounts for the fact that points in are located only on a hemisphere.

Estimate on K

Since HS_m is covered by a collection of θ-regular n-simplices, K can be estimated by taking the ratio between the volume of the sphere HS_m and the volume of a θ-regular n-simplex. Since θ is in general assumed to be small, we will approximate the volume of a θ-regular n-simplex on the sphere HS_m with the volume ω_n(θ) of a corresponding θ-regular n-simplex in the Euclidean space ℝⁿ:

$${\omega }_n(\theta )=\frac{\sqrt{n+1}}{n!\sqrt{2^n}}{\theta }^n.$$ (3.3)

It then follows that the number K of n-simplexes can be estimated as

$$K=\frac{{\mathbf{V}}_n}{{\omega }_n(\theta )},$$ (3.4)

where the volume of the sphere V_n is given by the formula [25]

$${\mathbf{V}}_n=\{\begin{array}{ll}\frac{{(2\pi )}^{(n+1)/2}}{2·4\cdots (n-1)}\hfill & \text{if}n\text{is}\text{odd};\hfill \\ \frac{2{(2\pi )}^{n/2}}{1·3\cdots (n-1)}\hfill & \text{if}n\text{is}\text{even}.\hfill \end{array}$$

Estimate on ν

For a typical vertex v in Δ( ), a small neighborhood around v in HS_m is covered by the θ-regular n-simplices having v as one of their vertices. Again, assuming θ is small, we can approximate this using Euclidean geometry, by transforming the neighborhood U onto the tangent space T_v at v using the log map. The geodesic ball B_θ of radius θ on HS_m is mapped to the Euclidean ball of radius θ and the image of each n-simplex under the log map can be approximated by a regular n-simplex in the Euclidean space with side length θ. See Figure 3.2. It follows that the degree of v can be estimated as the ratio between the volume of the unit n-dimensional ball and the volume of regular n-simplex in ℝⁿ with side length θ. The volume Vⁿ of an n-ball in ℝⁿ with radius r = 1 is given by the formula [25]

Fig. 3.2.

Left: For small θ, we can approximate the volume of a θ-regular spherical simplex by the volume of a θ-regular Euclidean simplex. The exponential map Exp_p maps a neighborhood of the origin in the tangent space T_p diffeomorphically onto a neighborhood at p. Since the derivative of Exp_p at p is the identity, for small enough θ, Exp_p is close to an isometry in B_θ. Right: The average degree of a vertex, ν, can be approximated by the number of θ-regular simplexes contained in the ball of radius θ.

$${\mathbf{V}}^n=\{\begin{array}{ll}\frac{{(2\pi )}^{n/2}}{2·4\cdots n}\hfill & \text{if}n\text{is}\text{even};\hfill \\ \frac{2{(2\pi )}^{(n-1)/2}}{1·3\cdots n}\hfill & \text{if}n\text{is}\text{odd}.\hfill \end{array}$$

The degree ν is then estimated as

$$\nu =\frac{{\mathbf{V}}^n}{{\omega }_n(1)}.$$ (3.5)

Combining Equations 3.3, 3.4, 3.5, we have

$$\begin{array}{l}\mathcal{N}(\epsilon ,m)=\#\text{of}\text{points}\text{in}\mathcal{C}\approx \frac{1}{2}\frac{(n+1){\mathbf{V}}_n}{{\omega }_n(\theta )\frac{{\mathbf{V}}^n}{{\omega }_n(1)}}=\frac{(n+1){\mathbf{V}}_n}{2{\mathbf{V}}^n{\theta }^n}\\ =\frac{(n+1){\mathbf{V}}_n}{2{\mathbf{V}}^n}{f}_{\eta }{(\epsilon )}^{-n}.\end{array}$$ (3.6)

In the remaining section, we will work out the implication of the above estimate for 2^nd, 4^th and 6^th-order tensors.

3.2. Second-Order Tensors

A quadratic homogeneous polynomial P(x, y, z) in ℝ³ has six coefficients P(x, y, z) = ax² + by² + cz² + dxy + exz + fyz. It can be written in a matrix form as,

$$P(x,y,z)=[\begin{array}{lll}x\hfill & y\hfill & z\hfill \end{array}]\left[\begin{array}{lll}a\hfill & \frac{d}{2}\hfill & \frac{e}{2}\hfill \\ \frac{d}{2}\hfill & b\hfill & \frac{f}{2}\hfill \\ \frac{e}{2}\hfill & \frac{f}{2}\hfill & c\hfill \end{array}\right]\left[\begin{array}{l}x\hfill \\ y\hfill \\ z\hfill \end{array}\right]={\mathbf{x}}^{\top }\mathbf{Sx}.$$

Positive semi-definiteness of the polynomial P(x, y, z) is equivalent to the positive semi-definiteness of the matrix S. It follows that determining positive semi-definiteness of a homogeneous quadratic polynomial is straightforward by examining eigenvalues of S: S is positive semi-definite if and only its eigenvalues λ₁, λ₂, λ₃ are all non-negative and S can be written as

$$\mathbf{S}={\lambda }_1{\mathbf{v}}_1^{\top }{\mathbf{v}}_1+{\lambda }_2{\mathbf{v}}_2^{\top }{\mathbf{v}}_2+{\lambda }_3{\mathbf{v}}_3^{\top }{\mathbf{v}}_3,$$

where v_i is the unit eigenvector with eigenvalue λ_i for i = 1, 2, 3. It follows that P(x, y, z) can be written as a sum of three linear polynomials p₁(x), p₂(x), p₃(x),

$$P(\mathbf{x})=p_1{(\mathbf{x})}^2+p_2{(\mathbf{x})}^2+p_3{(\mathbf{x})}^2,$$

with $p_i(\mathbf{x})=\sqrt{{\lambda }_i}{\mathbf{v}}^{\top }\mathbf{x}$.

With m = 1, the sphere HS_m has dimension n = 2. According to Proposition 2.3, the map ${\mathcal{F}}_1^2$ is isotropic with respect to the L¹-norm and η = 1. Equation 3.6 (together with Equation 3.2) then gives

$$N(\epsilon ,1)\approx \frac{3{\mathbf{V}}_2}{2{\mathbf{V}}^2}{\left(\frac{1}{\epsilon }\right)}^{\frac{2}{3}}=6{\left(\frac{1}{\epsilon }\right)}^{\frac{2}{3}}.$$ (3.7)

More Precise Estimate

For the linear case m = 1, since HS_m is the two-sphere S², its geometry is well-known and a better estimate on N can be obtained. Given θ, S² is covered by geodesic triangles whose sides have lengths of approximately θ. Approximating the areas of these geodesic triangles with the area of an Euclidean equilateral triangles with side θ gives ${\theta }^2\sqrt{3}/4$. Let F, E, V denote the number of triangles, edges and vertices in the triangulation Δ( ). According to Euler’s formula

$$F-E+V=\chi ({\mathbf{S}}^2)=2$$

where χ(S²) is the Euler characteristic of S². Since E = 3F/2, V = 2 + F/2 ≈ F/2. This gives ν = 6 as the average degree of a vertex on S². Our estimate on the degree ν in Equation 3.5 in this case gives $\nu =4\pi /\sqrt{3}\approx 7.2$, which gives a 20% overestimate.

The area A of a geodesic triangle on S² with three interior angles α, β, γ is given as [1]

$$A=\alpha +\beta +\gamma -\pi .$$

In particular, for a geodesic equilateral triangle on S² with side length θ, its angle α is given as

$$\alpha ={cos}^{-1}(\frac{cos\theta -{cos}^2\theta }{{sin}^2\theta }),$$

and the estimate on the number of triangles is

$$K=\frac{4\pi }{3{cos}^{-1}(\frac{cos\theta -{cos}^2\theta }{{sin}^2\theta })-\pi }.$$

Let $4tan\theta {sin}^2\frac{\theta }{2}=\epsilon$ and θ = f(ε) be the solution to the trigonometric equation. It then follows that

$$N(\epsilon ,1)=\frac{\pi }{3{cos}^{-1}(\frac{cos(f(\epsilon ))-{cos}^2(f(\epsilon ))}{{sin}^2(f(\epsilon ))})-\pi }.$$ (3.8)

In Figure 3.1, we compare the two estimates using Equations 3.8 and 3.7. For ε = 0.1, Equation 3.7 gives ≈ 30. And for ε = 0.01 and 0.001, it gives ≈ 130 and 600, respectively. As for Equation 3.8 it gives ≈ 34, 156, 725 for ε = 0.1, 0.01, 0.001, respectively.

3.3. Fourth-Order Tensors

In this case, m = 2 and H₂ and HS₂ have dimensions six and five, respectively. The map ${\mathcal{F}}_2^2$ is no longer isotropic with respect to L¹-norm in HS₂. An analytic evaluation of the matrix Λ² gives

$${\mathrm{\Lambda }}^2=\frac{4\pi }{5}\left(\begin{array}{cccccc}1& 1/3& 1/3& 0& 0& 0\\ 1/3& 1& 1/3& 0& 0& 0\\ 1/3& 1/3& 1& 0& 0& 0\\ 0& 0& 0& 1/3& 0& 0\\ 0& 0& 0& 0& 1/3& 0\\ 0& 0& 0& 0& 0& 1/3\end{array}\right).$$

The singular values of Λ² arranged in the descending order are

$$\sigma ({\mathrm{\Lambda }}^2)=\frac{4\pi }{3}[1,2/5,2/5,1/5,1/5,1/5].$$

This gives η = 5, and Equation 3.6 gives

$$N\approx \frac{3{\mathbf{V}}_5}{{\mathbf{V}}^5}{f}_{\eta =5}{(\epsilon )}^{-5}=\frac{90\pi }{16}{f}_{\eta =5}{(\epsilon )}^{-5}.$$

For ε = 0.1, this yields N ≈ 176790. However, in H₂, the polynomial v(x, y, z) = x²+y²+z² is the constant function 1 on S². In particular, $u(x,y,z)=v(x,y,z)/\sqrt{3}\in {\mathbf{HS}}_2$, and ${\left|\right|{\mathcal{F}}_2^2(u)\left|\right|}_1=4\pi /3$. The map ${\mathcal{F}}_2^2$ stretches the constant polynomial considerably more than any other quadratic polynomials, and this is the reason for the large condition number η. Let ℝu denote the one-dimensional subspace in H₂ spanned by the constant polynomial u(x, y, z), and W its orthogonal complement,

$${\mathbf{H}}_2=\mathbb{R}u\oplus \mathbf{W}.$$

The intersection of the sphere HS₂ with the subspace W is a four-sphere S⁴. If we specialize to this four-sphere, i.e., polynomials orthogonal to the constant polynomial x² + y² + z², the condition number η becomes 2 and the dimension of the sphere drops by one. Theorem 2.4 then provides the following estimate on the number of points

$$\mathcal{N}\approx \frac{5{\mathbf{V}}_4}{2{\mathbf{V}}^4}{f}_{\eta =2}{(\epsilon )}^{-4}=\frac{40}{3}{f}_{\eta =2}{(\epsilon )}^{-4}.$$

This number is considerably less than 176790. For example, for ε = 0.1, we have ≈ 1800 and for ε = 0.05, 0.01, ≈ 4670, 39620, respectively.

3.4. Sixth-Order Tensors

In this case, m = 3 and H₃, HS_m have dimensions 10, 9, respectively. The map ${\mathcal{F}}_3^2$ is again non-isotropic with respect to L¹-norm in H₆. The singular values of Λ³ arranged in the descending order are

$$\sigma ({\mathrm{\Lambda }}^3)=4\pi [\frac{19+\sqrt{193}}{210},\frac{19+\sqrt{193}}{210},\frac{19+\sqrt{193}}{210},\frac{19-\sqrt{193}}{210},\frac{19-\sqrt{193}}{210},\frac{19-\sqrt{193}}{210},2/105,2/105,2/105,1/105].$$

The condition number η = 16.44, which is quite substantial. However, similar analysis as above can be applied to eliminate polynomials in HS₃ coming from polynomials of lower degree to substantially decrease the condition number. First, the three linear polynomials x, y, z are now embedded in H₃ as x(x² + y² + z²), y(x² + y² + z²), z(x² + y² + z²). Let r̂(x), ŝ(x), t̂(x), r(x), s(x), t(x) be the following polynomials

$$\begin{array}{l}\widehat{r}(\mathbf{x})=x(x^2+y^2+z^2)/\sqrt{3},r(\mathbf{x})=0.7184x^3+0.3951\widehat{r}(\mathbf{x}),\\ \widehat{s}(\mathbf{x})=y(x^2+y^2+z^2)/\sqrt{3},s(\mathbf{x})=0.7184y^3+0.3951\widehat{s}(\mathbf{x}),\\ \widehat{t}(\mathbf{x})=z(x^2+y^2+z^2)/\sqrt{3},t(\mathbf{x})=0.7184z^3+0.3951\widehat{t}(\mathbf{x}).\end{array}$$

The three polynomials r(x), s(x), t(x) are responsible for the three largest singular values of Λ³. The smallest singular value of 4π/105 comes from the polynomial q(x) = xyz. Let W denote the six-dimensional subspace in H₃ that is the orthogonal complement of the subspace spanned by r(x), s(x), t(x) and q(x),

$${\mathbf{H}}_3=\mathbb{R}r\oplus \mathbb{R}s\oplus \mathbb{R}t\oplus \mathbb{R}q\oplus \mathbf{W}.$$

The sphere in W is five-dimensional, and the condition number of ${\mathcal{F}}_3^2$ on S⁵ is η = 1.2769.

$$\mathcal{N}\approx \frac{3{\mathbf{V}}_5}{{\mathbf{V}}^5}{f}_{\eta =1.27}{(\epsilon )}^{-5}=\frac{90\pi }{16}{f}_{\eta =1.27}{(\epsilon )}^{-5}.$$

For ε = 0.1, 0.05, 0.01, the result above gives ≈ 1943, 6021, 85495, respectively.

4. Experimental Results

In this section we experimentally validate the proposed theory and at the end of this section we present an application to Diffusion-Weighted MRI. In all the experiments we use tensors in ℝ³, which can be visualized by plotting the corresponding homogeneous polynomial P(x, y, z) as a spherical function (see Fig. 4.1). Such tensor glyphs can be generated by scaling the radius of a unit sphere at orientation x = [x y z]^T with the value of P(x, y, z). Additionally, we assign a color to each tensor glyph by using the following coloring scheme: we use the method in [11, 22] to compute the unit vector [x y z]^T that maximizes P(x, y, z) and then we assign to the R, G, B color channels the squares of the three components in the vector x (i.e. R = x², G = y², B = z²). This color map produces smooth color transitions when visualizing fields of tensors such as the diffusion tensor fields.

Fig. 4.1.

Examples of randomly computed symmetric positive semi-definite tensors in Ω₂, Ω₄, Ω₆. The tensor glyphs are shown.

First, we construct a dataset with samples from Ω_2m as follows: we first generate random vectors in ℝ^d(m) using the normal distribution N(μ = 0, σ² = 1) in d(m) = 3, 6, and 10 dimensions, and we use them as coefficients of linear, quadratic and cubic homogeneous polynomials p ∈ HS₁, HS₂, HS₃ in three variables, respectively. Then we construct 2^nd, 4^th and 6^th-order positive semi-definite tensors that belong to Ω_2m by taking sums of squares of the polynomials in HS₁, HS₂, HS₃, respectively. This process is repeated for 5000 times for each order, producing a dataset of 15000 tensors in total. Several of the generated tensors are shown in Fig. 4.1(right). The primary goal of the aforementioned process is to generate samples from Ω_2m in order to test the error analysis presented in Section 3, and it should not be perceived as a DW-MRI simulation as in this section we do not discuss any application of the proposed method to DW-MR imaging.

In order to investigate how many generators in the finitely-generated cone ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ are necessary for our algorithm to approximate accurately a set of given tensors, we apply our framework to the previously described synthetic dataset using finite subsets ∈ HS_m of various sizes N. The sets are constructed as the vertices computed by triangulating the unit n-sphere. The triangulation is based on a variation of the algorithm for mesh generation presented in [39], which extends to any dimension n of the n-sphere. This method is an iterative force-based technique that uses a force displacement function to move the nodes of the mesh and the Delaunay triangulation [14], which is a fundamental and widely used triangulation process, to adjust the topology (i.e. the edges). Obviously, in our particular case we discard the edge information since we only need the finite set of nodes. This algorithm produces at the end the finite subsets ∈ HS_m for different predefined sizes N.

We first use the constructed finite sets C in a numerical framework for approximating the error rate ε achieved by the finitely-generated cone ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ for m = 1, 2, 3. The numerical calculations were performed by randomly generating points in the n-sphere and testing if each point lies inside or outside the cone ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$. The error rate ε is the ratio of the points outside the cone over the total number of generated points. For each numerical computation we used 100k points. The numerical approximations are shown as circles in Fig. 4.2. By observing the figures we can see that in most of the cases the numerical approximations are close to the proposed formulas for computing N. We should note that the results are based on the computed sets using the method in [39]. One may expect that the results will be slightly different if another method is employed for triangulating the n-sphere.

Fig. 4.2.

Comparison of the proposed formulas for computing N for m=1,2,3 with results produced using a numerical approximation algotithm. The horizontal axis show the the accuracy achieved by N finite generators (vertical axis) in the unit n-sphere. The circles show the numerical results produced for specific sets of various sizes N.

We also use the sets in a non-negative least squares (NNLS) optimization framework [28] in order to estimate tensors from the finitely-generated cone ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ that approximate the given 15000 tensors. For each order of tensors, the NNLS system is formulated as Aw = b, where A a matrix constructed from , w the unknown solution vector and b contained the values of the given positive-semidefinite homogeneous polynomial at K = 81 three-dimensional unit vectors x₁ ··· x₈₁ (producing 81 components of b as b₁ = P (x₁) ··· b₈₁ = P (x₈₁)) for each tensor in the dataset. Although this problem seems extremely unconstrained in general, in our particular case the NNLS algorithm by definition constrains the number of non-zero elements in the solution vector to be at most d(2m), which is significantly smaller than the number of data points K in all of our experiments. In order to estimate such a constrained solution the NNLS algorithm implements a basis selection mechanism that starts with a set of possible basis vectors in , computes the associated dual vector, and then reselects the basis in the solution by iteratively performing swaps in order to minimize the entries in the dual vector until they are all non positive. In our particular case of m = 1, 2, 3 the estimated unknown non-zero entries are 6, 15, 28 respectively which are all significantly smaller than the number of given samples K = 81. For a detailed description of the NNLS algorithm the reader is referred to [28].

The solutions w provide tensors in ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$ that approximate the given tensors in Ω_2m, for m = 1, 2, and 3. The computed tensors are compared to the ground truth (given) tensors using the relative L¹-error (fitting error):

$$\frac{{\int }_{{\mathbf{S}}^2}\mid P_{\mathit{given}}(\mathbf{x})-P(\mathbf{x})\mid d\mathbf{x}}{{\int }_{{\mathbf{S}}^2}\mid P_{\mathit{given}}(\mathbf{x})\mid d\mathbf{x}}.$$ (4.1)

The histograms of the errors found in the experiments (measured by Eq. 4.1) are plotted in Fig. 4.3 for the case of 2^nd, 4^th, and 6^th-order tensors, respectively. Obviously, by increasing , i.e. the number of generators in the finitely-generated subcone ${\mathrm{\sum }}_{2m}^{\mathcal{C}}$, the error decreases correspondingly. The table in Fig. 4.3 reports the mean errors for various difference sizes N of the generator set.

Fig. 4.3.

Histograms of tensor fitting errors obtained by our method for the case of 2^nd, 4^th, and 6^th-order tensors respectively, using various sizes N of the set . The vertical axis corresponds to the percentage of the tensors in the dataset (i.e. number of occurances), and the horizontal axis corresponds to the given fitting error.

The experimental results presented in Fig. 4.3 and Fig. 4.4 validate empirically our method as the results corroborate well with our previous analysis on the number of generators required for a given relative error bound. For 2^nd-order tensors, the analysis in Section 3 shows that for the error to be less than ε = 10%, 1%, 0.1%, it requires approximately N ≈ 30, 130 and 600 generators, respectively. The first plot in Fig. 4.3 shows that with N = 45, there are no occurrences of error greater than 10%, and with N = 150, there are no occurrences of error greater than 1%. With 321 generators, the error becomes negligible. For 4^th-order tensors, our analysis shows that for the error to be less than ε = 10%, 5%, it requires approximately N ≈ 1800, 4670 generators, respectively. This can be seen from the second plot in Fig. 4.3. With N < 1500 generators, there are occurrences of 10% error, and with N ≥ 1500, there are no occurrences of error greater than 10%. To decrease the error under 5% level, the plot shows that we need at least N = 3000 generators. Finally, for 6^th-order tensors, our analysis shows that for the error to be less than ε = 10% and 5%, it requires approximately N ≈ 1943 and 6021 generators, respectively. The third plot in the figure show that at N = 3000, there is only a small percentage of errors greater than 10%, and with N = 6000, there is an even smaller percentage (less than 1%) of errors greater than 5%. In most cases, our earlier analysis underestimate the required numbers of generators, and this is not surprising as these analysis are themselves based on several approximations. Nevertheless, the experimental results do agree in general with the predictions made in Section 3.

Fig. 4.4.

Plots of the running time of our method for fitting one tensor versus the approximation error for the case of 2^nd, 4^th, and 6^th-order tensors, using various sizes N of the set . The vertical axis corresponds to the obtained mean fitting error, and the horizontal axis corresponds to the execution time.

Figure 4.4 shows the running time of the optimization method for fitting one tensor versus the approximation error for various orders and number of generators N in the set . The running times are measured using an Intel Pentium Dual CPU at 1.60 GHz and 1GB RAM. The plots demonstrate that the proposed technique can efficiently estimate positive tensors of various orders. More specifically, 2^nd, 4^th, and 6^th-order tensors can be estimated using finitely-generated subcones of size N = 45, N = 900, and N = 6000 at 0.5ms, 12ms, and 243ms, respectively.

4.1. Application: Diffusion-Weighted MRI

Finally, we present an application of the proposed tensor approximation theory to Diffusion-Weighted MRI (DW-MRI). In several DW-MRI processing methods, a diffusion tensor is computed from the acquired diffusion-weighted signals. Negative diffusion values are non-physical; therefore, appropriate methods such as our proposed framework are necessary to ensure positive semi-definiteness of the estimated Diffusion tensors.

In order to demonstrate the necessity for estimating tensors with the positivity constraints, we compare our method with an existing one that computes tensors without the constraints [35]. In this experiment, we use the aforementioned synthetic dataset of 6^th-order tensors, and we sample the corresponding homogeneous polynomials using K = 81 3-dimensional unit vectors x₁ ··· x₈₁ in the Stejskal-Tanner model [45], producing 81 DW-MRI samples for each tensor in the dataset. Various levels of Rician noise are added to the samples with standard deviations ranging from σ = 0.04 up to σ = 0.12. The noisy datasets are given as inputs to: a) the proposed algorithm (using N = 6000), and b) the method proposed in in [35], which is one of the several existing methods in the literature [15, 19] that estimate 6^th-order tensors. For both, the computed 6^th – order tensors P(x) are compared to the ground truth tensors using the error defined in Eq. 4.1.

Figure 4.5 shows the comparison of the fitting errors between the two methods for various levels of noise in the data. The results conclusively demonstrate that tensors estimated using positivity constraints approximate the data significantly better than the ones without. We also note that this result agrees with similar comparisons reported earlier for tensors of lower orders (e.g. 4^th-order comparison in [5]), showing that the errors incurred in approximating positive-valued functions are significantly smaller when positivity constraints are enforced in the process. Our current results have provided further evidence that supports the importance of imposing positivity constraints in this context.

Fig. 4.5.

Comparison of the 6^th-order tensor fitting errors obtained by the proposed method and the technique in [35] for various Rician noise levels in the data.

In order to illustrate the performance of our framework on real data sets, we applied the method to a DW-MRI data set of an excised rat hippocampus (shown in Fig. 4.6). The data set contains 46 images acquired using a pulsed gradient spin echo pulse sequence, with 45 different diffusion gradients and approximate b value of 1250s/mm². Figure 4.6 shows the computed 6^th-order diffusion tensor field. The highlighted regions of interest demonstrate the variability of the estimated structures. At each voxel, the fiber orientations can be estimated from the peaks of the displacement probability, which can be computed from the diffusion tensors as was shown in [5].

Fig. 4.6.

DW-MRI dataset from an isolated rat hippocampus. The image without diffusion weighting (S₀) is shown on the top left. The 6^th-order diffusion tensors estimated by the proposed method are shown as a field of spherical functions. The three regions of interest depict 6^th-order diffusion tensors that model one, two, and three fiber structures.

Finally, Fig. 4.7 presents the results obtained by applying our method to a DW-MRI dataset from an excised rat optic chiasm. The data acquisition protocol was the same as in the rat hippocampus dataset. The computed field of 4^th-order diffusion tensors is shown in the center. Using the estimated diffusion tensors, we can compute the underlying fiber orientations by computing the orientations that correspond to the maxima of the water molecule displacement probabilities. The computed fiber orientations are shown on the right and they agree with the known fiber orientations in the optic chiasm. Further quantitative validations of these orientations with respect to those from histology will be performed as part of our future work.

Fig. 4.7.

DW-MRI dataset from an isolated rat optic chiasm. A field of 4^th-order diffusion tensors computed by the proposed method is shown in the central plate. The corresponding estimated fiber orientations are shown on the right.

5. Discussion and conclusions

Symmetric positive semi-definite tensors have been used in many applications. Although there are existing methods for imposing positivity constraints on the estimated tensors of order two and four, none of these techniques can be easily extended to higher orders. In this paper, we presented a framework for estimating PSD tensors of any order by approximating the space (cone) of PSD tensors with a finitely-generated subcone Σ_2m. We discussed in detail the geometry of the higher-order tensors, and we presented an explicit characterization of the approximation, using the subset of semi-definite tensors that can be written as a sum of squares of tensors of order m. This approximation leads to a non-negative linear least-squares (NNLS) optimization problem, which can be efficiently solved, as it was demonstrated using synthetic datasets and real diffusion-weighted MR images.

An interesting property of the NNLS optimization algorithm is that it produces sparse solution vectors. In our particular case, although the problem seems significantly unconstrained, the solution vector contains at most d(2m) non-zero weights, which corresponds to the rank of the basis matrix. Therefore if the finitely-generated set contains a few thousands bases, the algorithm will select only 6, 15, 28 for tensors of order 2, 4, and 6 respectively. Note that the number of non-zero weights in the solution vector equals to the number of the unique unknown parameters of the symmetric tensor in each case. The sparsity of NNLS in comparison with other optimization techniques for modeling the diffusion-weighted MR signal has also been studied in [27].

In our experiments the sets were generated by tessellating the unit n-sphere using the iterative force-based technique in [39]. The vertices produced by this algorithm form the finite subset ∈ HS_m for different predefined sizes N. An alternative approach could involve constructing as a finite dictionary of elements in HS_m by running a training algorithm on a control dataset [31]. A finite set of diffusion basis for multi-fiber reconstruction is also employed by the method in [43].

One of the advantages of the proposed algorithm is that it enforces positive semi-definite constraints to the estimated tensors. The need for positivity constraints in DW-MRI has been demonstrated in [6] and [5]. It has been shown that unconstrained methods may yield negative diffusivities in real datasets, especially in voxels with high anisotropy or in the presence of noise in the data.

Finally, although high order tensors can approximate several distinct fiber orientations, in the current standard clinical settings for DW-MRI acquisition most of the multi-fiber reconstruction techniques cannot estimate more than two fiber orientations [41], due to the low diffusion weighting (b-value) and the small number of gradient orientations. However, theoretically or in experimental settings with higher b-values and larger sets of diffusion gradient orientations, the proposed technique can estimate up to 2 and 3 distinct fiber orientations using tensors of order 4 and 6 respectively, which also agrees with the results presented in [35].

Fig. 3.3.

The geometry of the map ${\mathcal{F}}_3^2$. Left: HS₃ is the nine-dimensional sphere S9. The decomposition of H₃ into four subspaces of dimensions of 3, 3, 3, 1 respectively implies that HS₃ contains separate copies of sphere S², S², S² and S⁰. ${\mathcal{F}}_3^2$ maps these spheres to spheres of radii 52π/83, 68π/699, 8π/105 and 4π/105, respectively. Right: The number of generators in Σ₂, Σ₄ and Σ₆ that can ensure the given accuracy requirement. The plots for m = 1, 2, 3 are in red, blue and green, respectively.