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As deep neural networks make their ways into different domains, their compute efficiency is becoming a first-order constraint. Deep quantization, which reduces the bitwidth of the operations (below 8 bits), offers a unique opportunity as it can reduce both the storage and compute requirements of the network super-linearly. However, if not employed with diligence, this can lead to significant accuracy loss. Due to the strong inter-dependence between layers and exhibiting different characteristics across the same network, choosing an optimal bitwidth per layer granularity is not a straight forward. As such, deep quantization opens a large hyper-parameter space, the exploration of which is a major challenge. We propose a novel sinusoidal regularization, called SINAREQ, for deep quantized training. Leveraging the sinusoidal properties, we seek to learn multiple quantization parameterization in conjunction during gradient-based training process. Specifically, we learn (i) a per-layer quantization bitwidth along with (ii) a scale factor through learning the period of the sinusoidal function. At the same time, we exploit the periodicity, differentiability, and the local convexity profile in sinusoidal functions to automatically propel (iii) network weights towards values quantized at levels that are jointly determined. We show how SINAREQ balance compute efficiency and accuracy, and provide a heterogeneous bitwidth assignment for quantization of a large variety of deep networks (AlexNet, CIFAR-10, MobileNet, ResNet-18, ResNet-20, SVHN, and VGG-11) that virtually preserves the accuracy. Furthermore, we carry out experimentation using fixed homogenous bitwidths with 3- to 5-bit assignment and show the versatility of SINAREQ in enhancing quantized training algorithms (DoReFa and WRPN) with about 4.8% accuracy improvements on average, and then outperforming multiple state-of-the-art techniques., Comment: Preliminary work. Under review

Patch priors have become an important component of image restoration. A powerful approach in this category of restoration algorithms is the popular Expected Patch Log-Likelihood (EPLL) algorithm. EPLL uses a Gaussian mixture model (GMM) prior learned on clean image patches as a way to regularize degraded patches. In this paper, we show that a generalized Gaussian mixture model (GGMM) captures the underlying distribution of patches better than a GMM. Even though GGMM is a powerful prior to combine with EPLL, the non-Gaussianity of its components presents major challenges to be applied to a computationally intensive process of image restoration. Specifically, each patch has to undergo a patch classification step and a shrinkage step. These two steps can be efficiently solved with a GMM prior but are computationally impractical when using a GGMM prior. In this paper, we provide approximations and computational recipes for fast evaluation of these two steps, so that EPLL can embed a GGMM prior on an image with more than tens of thousands of patches. Our main contribution is to analyze the accuracy of our approximations based on thorough theoretical analysis. Our evaluations indicate that the GGMM prior is consistently a better fit formodeling image patch distribution and performs better on average in image denoising task.

In this paper, we formulate the problem of inferring a Finite Mixture Model from discrete data as an optimal transport problem with entropic regularization of parameter $\lambda\geq 0$. Our method unifies hard and soft clustering, the Expectation-Maximization (EM) algorithm being exactly recovered for $\lambda=1$. The family of clustering algorithm we propose rely on the resolution of nonconvex problems using alternating minimization. We study the convergence property of our generalized $\lambda-$EM algorithms and show that each step in the minimization process has a closed form solution when inferring finite mixture models of exponential families. Experiments highlight the benefits of taking a parameter $\lambda>1$ to improve the inference performance and $\lambda\to 0$ for classification.

We focus on the maximum regularization parameter for anisotropic total-variation denoising. It corresponds to the minimum value of the regularization parameter above which the solution remains constant. While this value is well know for the Lasso, such a critical value has not been investigated in details for the total-variation. Though, it is of importance when tuning the regularization parameter as it allows fixing an upper-bound on the grid for which the optimal parameter is sought. We establish a closed form expression for the one-dimensional case, as well as an upper-bound for the two-dimensional case, that appears reasonably tight in practice. This problem is directly linked to the computation of the pseudo-inverse of the divergence, which can be quickly obtained by performing convolutions in the Fourier domain.

In this paper, we develop an approach to recursively estimate the quadratic risk for matrix recovery problems regularized with spectral functions. Toward this end, in the spirit of the SURE theory, a key step is to compute the (weak) derivative and divergence of a solution with respect to the observations. As such a solution is not available in closed form, but rather through a proximal splitting algorithm, we propose to recursively compute the divergence from the sequence of iterates. A second challenge that we unlocked is the computation of the (weak) derivative of the proximity operator of a spectral function. To show the potential applicability of our approach, we exemplify it on a matrix completion problem to objectively and automatically select the regularization parameter., Comment: This version is an update of our original paper presented at ICML'2012 workshop on Sparsity, Dictionaries and Projections in Machine Learning and Signal Processing