Algorithm 1.

The framework of online dictionary learning in mini-batch mode.

Input: Initialized dictionary D0 ∈ Rn×k, input data yi ∈ Rn, number of iterations T, regularization parameter λ ∈ R.

Output: Learned dictionary DT

A0 = 0, B0 = 0.

for t = 1 → T do

Randomly draw a set from Y : yt,1, yt,2, …, yt,τ.

for i = 1 → τ do

Sparse coding: ${\alpha }_{t,i}=\underset{\alpha \in {\mathbf{R}}^k}{argmin}\frac{1}{2}{\Vert y_{t,i}-D_{t-1}\alpha \Vert }_2^2+\lambda {\Vert \alpha \Vert }_1$.

end for

$A_t=\beta A_{t-1}+{\sum }_{i=1}^{\tau }{\alpha }_{t,i}{\alpha }_{t,i}^T,B_t=\beta B_{t-1}+{\sum }_{i=1}^{\tau }{y}_{t,i}{\alpha }_{t,i}^T$,

where $\beta =\frac{\theta +1-\tau }{\theta +1}$, and θ = tτ if t < τ, θ = τ2 + t − τ otherwise.

Dictionary update: Compute Dt, so that:

$\underset{D}{argmin}\frac{1}{t}{\sigma }_{i=1}^t\frac{1}{2}{\Vert y_i-D{\alpha }_i\Vert }_2^2+\lambda {\Vert {\alpha }_i\Vert }_1=\underset{D}{argmin}\frac{1}{t}(\frac{1}{2}Tr(D^T{DA}_t)-Tr(D^T{B}_t))$.

end for