On an adaptive regularization for ill-posed nonlinear systems and its trust-region implementation

In this paper we address the stable numerical solution of nonlinear ill-posed systems by a trust-region method. We show that an appropriate choice of the trust-region radius gives rise to a procedure that has the potential to approach a solution of the unperturbed system. This regularizing property is shown theoretically and validated numerically.Comment: arXiv admin note: text overlap with arXiv:1410.278

Efficient Identification of Butterfly Sparse Matrix Factorizations

Fast transforms correspond to factorizations of the form $\mathbf{Z} = \mathbf{X}^{(1)} \ldots \mathbf{X}^{(J)}$, where each factor $\mathbf{X}^{(\ell)}$ is sparse and possibly structured. This paper investigates essential uniqueness of such factorizations, i.e., uniqueness up to unavoidable scaling ambiguities. Our main contribution is to prove that any $N \times N$ matrix having the so-called butterfly structure admits an essentially unique factorization into $J$ butterfly factors (where $N = 2^{J}$), and that the factors can be recovered by a hierarchical factorization method, which consists in recursively factorizing the considered matrix into two factors. This hierarchical identifiability property relies on a simple identifiability condition in the two-layer and fixed-support setting. This approach contrasts with existing ones that fit the product of butterfly factors to a given matrix via gradient descent. The proposed method can be applied in particular to retrieve the factorization of the Hadamard or the discrete Fourier transform matrices of size $N=2^J$. Computing such factorizations costs $\mathcal{O}(N^{2})$, which is of the order of dense matrix-vector multiplication, while the obtained factorizations enable fast $\mathcal{O}(N \log N)$ matrix-vector multiplications and have the potential to be applied to compress deep neural networks

Approximation speed of quantized vs. unquantized ReLU neural networks and beyond

We deal with two complementary questions about approximation properties of ReLU networks. First, we study how the uniform quantization of ReLU networks with real-valued weights impacts their approximation properties. We establish an upper-bound on the minimal number of bits per coordinate needed for uniformly quantized ReLU networks to keep the same polynomial asymptotic approximation speeds as unquantized ones. We also characterize the error of nearest-neighbour uniform quantization of ReLU networks. This is achieved using a new lower-bound on the Lipschitz constant of the map that associates the parameters of ReLU networks to their realization, and an upper-bound generalizing classical results. Second, we investigate when ReLU networks can be expected, or not, to have better approximation properties than other classical approximation families. Indeed, several approximation families share the following common limitation: their polynomial asymptotic approximation speed of any set is bounded from above by the encoding speed of this set. We introduce a new abstract property of approximation families, called infinite-encodability, which implies this upper-bound. Many classical approximation families, defined with dictionaries or ReLU networks, are shown to be infinite-encodable. This unifies and generalizes several situations where this upper-bound is known

IML FISTA: Inexact MuLtilevel FISTA for Image Restoration

This paper presents IML FISTA, a multilevel inertial and inexact forward-backward algorithm, based on the use of the Moreau envelope to build efficient and useful coarse corrections. Such construction is provided for a broad class of composite optimization problems with proximable functions. This approach is supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We evaluate our approach on several image reconstruction problems and we show that it considerably accelerates the convergence of classical methods such as FISTA, for large-scale images

Efficient Identification of Butterfly Sparse Matrix Factorizations

International audienceFast transforms correspond to factorizations of the form $\mathbf{Z} = \mathbf{X}^{(1)} \ldots \mathbf{X}^{(J)}$, where each factor $\mathbf{X}^{(\ell)}$ is sparse and possibly structured. This paper investigates essential uniqueness of such factorizations, i.e., uniqueness up to unavoidable scaling ambiguities. Our main contribution is to prove that any $N \times N$ matrix having the so-called butterfly structure admits an essentially unique factorization into $J$ butterfly factors (where $N = 2^{J}$), and that the factors can be recovered by a hierarchical factorization method, which consists in recursively factorizing the considered matrix into two factors. This hierarchical identifiability property relies on a simple identifiability condition in the two-layer and fixed-support setting. This approach contrasts with existing ones that fit the product of butterfly factors to a given matrix via gradient descent. The proposed method can be applied in particular to retrieve the factorization of the Hadamard or the discrete Fourier transform matrices of size $N=2^J$. Computing such factorizations costs $\mathcal{O}(N^{2})$, which is of the order of dense matrix-vector multiplication, while the obtained factorizations enable fast $\mathcal{O}(N \log N)$ matrix-vector multiplications and have the potential to be applied to compress deep neural networks

Identifiability in Two-Layer Sparse Matrix Factorization

Sparse matrix factorization is the problem of approximating a matrix $\mathbf{Z}$ by a product of $J$ sparse factors $\mathbf{X}^{(J)} \mathbf{X}^{(J-1)} \ldots \mathbf{X}^{(1)}$. This paper focuses on identifiability issues that appear in this problem, in view of better understanding under which sparsity constraints the problem is well-posed. We give conditions under which the problem of factorizing a matrix into \emph{two} sparse factors admits a unique solution, up to unavoidable permutation and scaling equivalences. Our general framework considers an arbitrary family of prescribed sparsity patterns, allowing us to capture more structured notions of sparsity than simply the count of nonzero entries. These conditions are shown to be related to essential uniqueness of exact matrix decomposition into a sum of rank-one matrices, with structured sparsity constraints. In particular, in the case of fixed-support sparse matrix factorization, we give a general sufficient condition for identifiability based on rank-one matrix completability, and we derive from it a completion algorithm that can verify if this sufficient condition is satisfied, and recover the entries in the two sparse factors if this is the case. A companion paper further exploits these conditions to derive identifiability properties and theoretically sound factorization methods for multi-layer sparse matrix factorization with support constraints associated to some well-known fast transforms such as the Hadamard or the Discrete Fourier Transforms

Méthodes proximales multi-niveaux pour la restauration d'images

International audienceCet article présente une nouvelle méthode pour mettre en œuvre un algorithme forward-backward multi-niveau. En utilisant l’enveloppe de Moreau pour construire la correction apportée par les modèles grossiers, facilement calculable lorsque l’on connait sous forme explicite l’opérateur proximal des fonctions considérées, nous reformulons les algorithmes proximaux multi-niveaux précédemment introduits dans la littérature sous une forme plus simple, tout en en conservant les performances. Nous montrons la convergence des itérées vers un minimum dans le cas convexe, résultat fondamental pour des problèmes mal posés. Nous validons l’approche sur des problèmes de restauration d’image de grande taille

Optimal quantization of rank-one matrices in floating-point arithmetic---with applications to butterfly factorizations

We consider the problem of optimally quantizing rank-one matrices to low precision floating-point arithmetic. We first explain that the naive strategy of separately quantizing the two rank-one factors can be far from optimal, and we provide worst case error bounds to support this observation. We characterize the optimal solution as the quantization of suitably scaled factors of the rank-one matrix and we develop an algorithm of tractable complexity to find the optimal scaling parameters. Using random rank-one matrices, we show experimentally that our algorithm can significantly reduce the quantization error. We then apply this algorithm to the quantization of butterfly factorizations, a fundamental tool that appears in many fast linear transforms. We show how the properties of butterfly supports can be exploited to approach the problem via a series of rank-one quantization problems and we employ our algorithm as a building block in a heuristic procedure to quantize a product of butterfly factors. We show that, despite being only heuristic, this strategy can be much more accurate than quantizing each factor independently or, equivalently, can achieve storage reductions of up to 30% with no loss of accuracy