Discrete p-bilaplacian Operators on Graphs

Abstract

In this paper, we first introduce a new family of operators on weighted graphs called p-bilaplacian operators, which are the analogue on graphs of the continuous p-bilaplacian operators. We then turn to study regularized variational and boundary value problems associated to these operators. For instance, we study their well-posedness (existence and uniqueness). We also develop proximal splitting algorithms to solve these problems. We finally report numerical experiments to support our findings.

Introduction

Regularized variational problems and partial differential equations (PDEs) play an important role in mathematical modeling throughout applied and natural sciences. For instance, many variational problems and PDEs have been studied to model and solve important problems in a variety of areas such as, e.g., in physics, economy, data processing, computer vision. In particular they have been very successful in image and signal processing to solve a wide spectrum of applications such as isotropic and anisotropic filtering, inpainting or segmentation.

In many real-world problems, such as in machine learning and mathematical image processing, the data is discrete, and graphs constitute a natural structure suited to their representation. Each vertex of the graph corresponds to a datum, and the edges encode the pairwise relationships or similarities among the data. For the particular case of images, pixels (represented by nodes) have a specific organization expressed by their spatial connectivity. Therefore, a typical graph used to represent images is a grid graph. For the case of unorganized data such as point clouds, a graph can also be built by modeling neighborhood relationships between the data elements. For these reasons, there has been recently a wave of interest in adapting and solving nonlocal variational problems and PDEs on data which is represented by arbitrary graphs and networks. Using this framework, problems are directly expressed in a discrete setting where an appropriate discrete differential calculus can be proposed; see e.g., [4, 5] and references therein. This mimetic approach consists of replacing continuous differential operators, e.g., gradient or divergence, by reasonable discrete analogues, which makes it possible to transfer many important tools and results from the continuous setting.

Contributions. In this work, we introduce a novel class of p-bilaplacian operators on weighted graphs, which can be seen as proper discretizations on graphs of the classical p-bilaplacian operators [9]. Building upon this definition, we study a corresponding regularized variational problem as well as a boundary value problem. The latter naturally gives rise to p-biharmonic functions on graphs and equivalent definitions of p-biharmonicity [8]. For these two problems, we start by establishing their well-posedness (existence and uniqueness). We then turn to developing proximal splitting algorithms to solve them, appealing to sophisticated tools from non-smooth optimization. Numerical results are reported to support the viability of our approach.

Notations and Preliminary Results

Throughout this paper, we assume that is a finite connected undirected weighted graph without loops and parallel edges, where V is the set of vertices, E is the set of edges, and the symmetric function is the weight function. We denote by the edge that connects the vertices x and y, and we write for two adjacent vertices. For two vertices with we set and thus the set of edges E can be characterized by the support of the weight function , i.e., .

Let be the vector space of real-valued functions on the vertices of the graph. For a function the -norm of u is given by

We define in a similar way as the vector space of all real-valued functions on the edges of the graph.

Let and . The (nonlocal) gradient operator is defined as

This is a linear antisymmetric operator whose adjoint is the (nonlocal) weighted divergence operator denoted . It is easy to show that

For , the anisotropic graph p-Laplacian operator is thus defined by

Unless stated otherwise, in the rest of the paper, we assume .

p-biharmonic Functions on Graphs

We define p-biharmonic functions on graphs inspired by the way p-harmonic functions were introduced in [8] for networks. Let’s consider the following functional

1

Observe that is the standard Laplacian on a graphs, which is a self-adjoint operator.

Definition 1

We define the p-bilaplacian operator for a function by

Definition 2

Let . We say that a function u is p-biharmonic on A if it is a minimiser of the functional among functions in V with the same values in , that is, if

for every function , with in .

Inspired by [8], existence and uniqueness of p-biharmonic functions can be established using standard arguments.

Proposition 1

Let A subset of V and . The following assertions are equivalent:

• (i)the function u is p-biharmonic on A.
• (ii)
  the function u satisfies

  2

  for every function , with in .
• (iii)
  the function u solves

  

p-bilaplacian Dirichlet Problem on Graphs

Consider the following boundary value (Dirichlet) problem

3

where , , and . Observe that since the graph G is connected, there always exists a path connecting any pair vertices in . Our goal now is to establish well-posedness of (3). This will be derived using Dirichlet’s variational principe (hence the subscript d in ), which, in view of Proposition 1, amounts to equivalently studying the minimization problem

4

where is the subspace of the functions with a zero “trace”.

Theorem 1

The problem (3) has a unique solution.

Proof

Let be the indicator function of , i.e. it is 0 on and otherwise. By the Poincaré-type inequality established in Lemma 1, we get that is coercive. Since this objective is lower semicontinuous (lsc) by closedness of and continuity of , (4) has a minimizer. This together with strict convexity of on (see Lemma 2) then entails uniqueness.

Lemma 1

There is such that

5

for all . Thus is coercive on .

Proof

Set

Since the graph G is connected, there is such that forms a partition of V. By Jensen’s inequality, we have

, where and . Since on and forms a partition of V, it is easy to see that there exists such that

We arrive at the coercivity result by taking .

Let . We then have from Hölder and Young inequalities

where . Since the norms are equivalent in any finite-dimensional vector space, there exists with , such that

Choosing , , we get

whence coercivity of follows immediately.

Lemma 2

The functional is strictly convex on .

Proof

Assume that is not strictly convex on . Then there exist with such that for all . But since the function is strictly convex on for , this equality entails that on V, hence on A. Clearly satisfies

But we know from [8, Theorem 3.11 and Corollary 3.16] that on V, i.e., on V, leading to a contradiction.

p-bilaplacian Variational Problem on Graphs

In this section, we consider the following minimization problem, which is valid for any 1,

6

where is a linear operator, , is the regularization parameter, and is given by (1). Problems of the form (6) can be of great interest for graph-based regularization in machine learning and inverse problems in imaging; see [7] and references therein. Problem (6) is well-posed under standard assumptions.

Theorem 2

The set of minimizers of is non-empty and compact if and only if . If, moreover, either A is injective or , then has a unique minimizer.

Proof

For any proper lsc convex function f, recall its recession function from [11, Chapter 2], denoted . We have from the calculus rules in [11, Chapter 2] that

Since and are non-negative and coercive, we have from [11, Proposition 3.1.2] that their recession functions are positive for any non-zero argument. Equivalently,

Thus for all if and only if . Equivalence with the existence and compactness assertion follows from [11, Proposition 3.1.3].

Let’s turn to uniqueness. When A is injective, the claim follows from strict (in fact strong convexity) of the data fidelity term. Suppose now that . By strict convexity of and , a standard contradiction argument shows that for any pair of minimizers and , we have . This yields the uniqueness claim under the stated assumption.

Algorithms and Numerical Results

To solve both (3) and (6), we adopt a primal-dual proximal splitting (PDS) framework with an appropriate splitting of the functions and linear operators.

A PDS for the Boundary Value Problem (3)

Problem (3) is equivalent to (4). The latter takes the form

7

The latter can be solved with the following PDS iterative scheme [3], which reads in this case

8

where , is the orthogonal projector on the subspace (which has a trivial closed form), , and is the proximal mapping of the proper lsc convex function . The latter can be computed easily, see  [7] for details. Combining [3, Theorem 1], Proposition 1 and Theorem 1, the convergence guarantees of (8) are summarized in the following proposition.

Proposition 2

If , then the sequence provided by (8) converges to , where is a solution to (3), which is unique if .

A PDS for the Variational Problem (6)

For simplicity and space limitation, we restrict ourselves here to the case where A is the identity. In this case, inspired by the work in [6], we use the (accelerated) FISTA iterative scheme [2, 10] to solve the Fenchel-Rockafellar dual problem of (6), and recover the primal solution by standard extremality relationships. Our scheme reads in this case

9

where , .

Combining Theorem 2, [6, Theorem 2], [1, Theorem 1.1], the scheme (9) has the following convergence guarantees.

Proposition 3

The sequence converges to , the unique minimizer of (6), at the rate .

Numerical Results

We apply the scheme (9) to solve (6) in order to denoise a function f defined on a 2-D point cloud. We apply (3) in a semisupervised classification problem which amounts to finding the missing labels of a label function defined on a 2-D point cloud. The nodes of the graph are the points in the 2-D cloud and u(x) is the value at a point/vertex x. We choose the nearest neighbour graph with the standard weighting kernel when and 0 otherwise, where and are the 2-D spatial coordinates of the points in the cloud. The original point cloud used in our numerical experiments consists of points that are not on a regular grid. For the variational problem, the noisy observation is generated by adding a white Gaussian noise of standard deviation 0.5 to the original data, see Fig. 1(a). For the Dirichlet problem, the initial label function takes the values of the original data on a set of points/vertices where this set corresponds to the boundary data, it is chosen randomly and is equal to N/4 of the original points/vertices, see Fig. 1(b).

Fig. 1.

(a): results for denoising with . (b): results for a semisupervised classification problem with . For each setting, we show the original function on the point cloud, its observed version, and the result provided by each of our algorithms.