Publications by Shayan Aziznejad

Linear inverse problems with Hessian-Schatten total variation

In this paper, we characterize the class of extremal points of the unit ball of the Hessian-Schatten total variation (HTV) functional. The underlying motivation for our work stems from a general representer theorem that characterizes the solution set of re ...

Springer Heidelberg2024

Measuring Complexity of Learning Schemes Using Hessian-Schatten Total Variation

Michaël Unser, Joaquim Gonçalves Garcia Barreto Campos, Shayan Aziznejad

2023

Learning of Continuous and Piecewise-Linear Functions With Hessian Total-Variation Regularization

Michaël Unser, Joaquim Gonçalves Garcia Barreto Campos, Shayan Aziznejad

We develop a novel 2D functional learning framework that employs a sparsity-promoting regularization based on second-order derivatives. Motivated by the nature of the regularizer, we restrict the search space to the span of piecewise-linear box splines shi ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

Sparsest Univariate Learning Models Under Lipschitz Constraint

Michaël Unser, Shayan Aziznejad, Thomas Jean Debarre

Beside the minimizationof the prediction error, two of the most desirable properties of a regression scheme are stability and interpretability. Driven by these principles, we propose continuous-domain formulations for one-dimensional regression problems. I ...

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

Optimization Over Banach Spaces: A Unified View on Supervised Learning and Inverse Problems

Shayan Aziznejad

In this thesis, we reveal that supervised learning and inverse problems share similar mathematical foundations. Consequently, we are able to present a unified variational view of these tasks that we formulate as optimization problems posed over infinite-di ...

EPFL2022

The Wavelet Compressibility of Compound Poisson Processes

Julien René Pierre Fageot, Shayan Aziznejad

In this paper, we precisely quantify the wavelet compressibility of compound Poisson processes. To that end, we expand the given random process over the Haar wavelet basis and we analyse its asymptotic approximation properties. By only considering the nonz ...

2022

Coupled Splines for Sparse Curve Fitting

Michaël Unser, Shayan Aziznejad, Thomas Jean Debarre

We formulate as an inverse problem the construction of sparse parametric continuous curve models that fit a sequence of contour points. Our prior is incorporated as a regularization term that encourages rotation invariance and sparsity. We prove that an op ...

2022

Convex optimization in sums of Banach spaces

Michaël Unser, Shayan Aziznejad

We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a finite sum of compo ...

2021

Multikernel Regression with Sparsity Constraint

Michaël Unser, Shayan Aziznejad

In this paper, we provide a Banach-space formulation of supervised learning with generalized total-variation (gTV) regularization. We identify the class of kernel functions that are admissible in this framework. Then, we propose a variation of supervised l ...

2021

Graphic: Graph-Based Hierarchical Clustering For Single-Molecule Localization Microscopy

Michaël Unser, Daniel Sage, Shayan Aziznejad

We propose a novel method for the clustering of point-cloud data that originate from single-molecule localization microscopy (SMLM). Our scheme has the ability to infer a hierarchical structure from the data. It takes a particular relevance when quantitati ...

IEEE2021

Continuous-Domain Formulation of Inverse Problems for Composite Sparse-Plus-Smooth Signals

Michaël Unser, Shayan Aziznejad, Thomas Jean Debarre

We present a novel framework for the reconstruction of 1D composite signals assumed to be a mixture of two additive components, one sparse and the other smooth, given a finite number of linear measurements. We formulate the reconstruction problem as a cont ...

2021

Duality Mapping for Schatten Matrix Norms

Michaël Unser, Shayan Aziznejad

In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Holder inequality for Schatten norms. We prove in our main result that ...

2021

Shortest-support multi-spline bases for generalized sampling

Michaël Unser, Shayan Aziznejad, Alireza Naderi, Alexis Marie Frederic Goujon

Generalized sampling consists in the recovery of a function f, from the samples of the responses of a collection of linear shift-invariant systems to the input f . The reconstructed function is typically a member of a finitely generated integer-shift invar ...

2021

Wavelet analysis of the Besov regularity of Levy white noise

Julien René Pierre Fageot, Shayan Aziznejad

We characterize the local smoothness and the asymptotic growth rate of the Levy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothn ...

2020

Dictionary Learning with Statistical Sparsity in the Presence of Noise

Michaël Unser, Shayan Aziznejad, Emmanuel Emilien Louis Soubies

We consider a new stochastic formulation of sparse representations that is based on the family of symmetric alpha-stable (S alpha S) distributions. Within this framework, we develop a novel dictionary-learning algorithm that involves a new estimation techn ...

IEEE2020

Deep Neural Networks With Trainable Activations and Controlled Lipschitz Constant

Michaël Unser, Harshit Gupta, Joaquim Gonçalves Garcia Barreto Campos, Shayan Aziznejad

We introduce a variational framework to learn the activation functions of deep neural networks. Our aim is to increase the capacity of the network while controlling an upper-bound of the actual Lipschitz constant of the input-output relation. To that end, ...

2020

Support and approximation properties of Hermite splines

Michaël Unser, Virginie Sophie Uhlmann, Julien René Pierre Fageot, Shayan Aziznejad

In this paper, we formally investigate two mathematical aspects of Hermite splines that are relevant to practical applications. We first demonstrate that Hermite splines are maximally localized, in the sense that the size of their support is minimal among ...

2020

Learning Activation Functions in Deep (Spline) Neural Networks

Michaël Unser, Harshit Gupta, Joaquim Gonçalves Garcia Barreto Campos, Shayan Aziznejad, Pakshal Narendra Bohra

We develop an efficient computational solution to train deep neural networks (DNN) with free-form activation functions. To make the problem well-posed, we augment the cost functional of the DNN by adding an appropriate shape regularization: the sum of the ...

2020

Generating Sparse Stochastic Processes Using Matched Splines

Michaël Unser, Shayan Aziznejad, Leello Tadesse Dadi

We provide an algorithm to generate trajectories of sparse stochastic processes that are solutions of linear ordinary differential equations driven by Levy white noises. A recent paper showed that these processes are limits in law of generalized compound-P ...

2020

Sparse Dictionaries for Continuous-Domain Inverse Problems

Michaël Unser, Shayan Aziznejad, Thomas Jean Debarre

We study 1D continuous-domain inverse problems for multicomponent signals. The prior assumption on these signals is that each component is sparse in a different dictionary specified by a regularization operators. We introduce a hybrid regularization functi ...

2019