视频前背景分离论文之（4） A Probabilistic Approach to Robust Matrix Factorization

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本文介绍了概率正则化矩阵分解（PRMF）模型的基本原理及应用。PRMF模型结合了Laplace误差和Gaussian先验，对应于l1损失和l2正则化。文章深入探讨了模型的概率解释、重新表述以及优化过程。

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1、Abstract

PRMF：formulated with a Laplace error and a Gaussian prior which correspond to an l1loss and an l2regularizer.

2、MF mode

min U \in R m \times r, V \in R n \times r ∥ W ⊙ (Y - U V T) ∥ a a + λ u 2 ∥ U ∥ 22 + λ v 2 ∥ V ∥ 22

$\min_{U\in \mathbb R^{m \times r},V\in \mathbf R^{n\times r}} \Vert W\odot (Y-UV^T) \Vert_a^a+\frac{\lambda_u}{2}\Vert U\Vert_2^2+\frac{\lambda_v}{2}\Vert V\Vert_2^2$

3、PRMF

3.1 Probabilistic Interpretation

Probabilistic Interpretation：corresponding to a MAP estimation problem.

Y u i j | λ u v i j | λ v = = = U V T + E N (u i j | 0, λ - 1 u) N (v i j | 0, λ - 1 v)

$\begin{aligned} Y &=& UV^T+E \\ u_{ij}|\lambda_u &=& N( u_{ij}| 0,\lambda_u^{-1}) \\ v_{ij}|\lambda_v &=& N( v_{ij}| 0,\lambda_v^{-1}) \end{aligned}$
suppose

eij $e_{ij}$ is sampled from Laplace distribution

L(eij|0,λ) $L(e_{ij}|0,\lambda)$ :

p (E | λ) = (λ 2) m n exp {- λ ∥ E ∥ 1}

$p(E|\lambda) = (\frac{\lambda}{2})^{mn}\exp\{-\lambda\Vert E\Vert_1\}$
from Bayes’,

log p (U, V | Y, λ, λ u, λ v) = - λ ∥ Y - U V T ∥ a a + λ u 2 ∥ U ∥ 22 + λ v 2 ∥ V ∥ 22

$\log p(U,V|Y,\lambda,\lambda_u,\lambda_v) = -\lambda\Vert Y-UV^T \Vert_a^a+\frac{\lambda_u}{2}\Vert U\Vert_2^2+\frac{\lambda_v}{2}\Vert V\Vert_2^2$

3.2 Model Reformulation

Laplace distribution:

p (z | u, α 2) = α 2 2 exp (- α 2 | z - u |)

$p(z|u,\alpha^2)=\frac{\alpha^2}{2}\exp(-\alpha^2|z-u|)$ is equivalently expressed as a scaled mixture of Gaussians:

L (z | u, α 2) = \int \infty 0 N (z | u, τ) E x p o n (τ | α 2) d τ

$L(z|u,\alpha^2)=\int_0^\infty N(z|u,\tau)\mathrm {Expon}(\tau|\alpha^2)\mathrm d\tau$
where

Expon(τ|α2) $\mathrm{Expon}(\tau|\alpha^2)$ is:

p (τ | α 2) = α 2 2 exp (- α 2 τ 2)

$p(\tau|\alpha^2)=\frac{\alpha^2}{2}\exp(-\frac{\alpha^2\tau}{2})$

Matrix $T = [\tau_{ij} ]\in \mathbb R^{m\times n}$ , where each element $\tau_{ij}$ is a latent variable with exponential prior for the corresponding $y_{ij}$ .

y i j | U, V, T u i j | λ u v i j | λ v τ i j | λ \sim \sim \sim \sim N (y i j | u T i v j, τ i j) N (u i j | 0, λ - 1 u) N (v i j | 0, λ - 1 v) E x p o n (τ i j | λ / 2)

$\begin{aligned} y_{ij}|U,V,T&\sim& N(y_{ij}|\mathbf u_i^T\mathbf v_j,\tau_{ij}) \\ \mathbf u_{ij}|\lambda_u&\sim& N(\mathbf u_{ij}|\mathbf 0,\lambda_u^{-1}) \\ \mathbf v_{ij}|\lambda_v&\sim& N(\mathbf v_{ij}|\mathbf 0,\lambda_v^{-1}) \\ \tau_{ij}|\lambda &\sim&\mathrm Expon(\tau_{ij}|\lambda/2) \end{aligned}$