arXiv subjects

Shayan Aziznejad

Publications and source records attributed to Shayan Aziznejad.

15 recordsLinked to original sources

arXiv2026

EvoFlows: Evolutionary Edit-Based Flow-Matching for Protein Engineering

We introduce EvoFlows, a variable-length protein sequence-to-sequence modeling approach designed for protein engineering. Existing protein language models are poorly suited for optimization tasks: autoregressive models require full sequence generation, masked language and discrete diffusion models rely on pre-specified mutation locations, and no existing methods naturally support insertions and deletions relative to a template sequence. EvoFlows learns mutational trajectories between evolutionarily related protein sequences via edit flows, allowing it to perform a controllable number of mutations (insertions, deletions, and substitutions) on a template sequence, predicting not only _which_ mutation to perform, but also _where_ it should occur. Through extensive _in silico_ evaluation on diverse protein families from UniRef and OAS, we show that EvoFlows generates variants that remain consistent with natural protein families while exploring farther from template sequences than leading baselines.

cs.LG↗

arXiv2022

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 regularized linear inverse problems in terms of the extremal points of the regularization ball. Our analysis is mainly based on studying the class of continuous and piecewise linear (CPWL) functions. In particular, we show that in dimension $d=2$, CPWL functions are dense in the unit ball of the HTV functional. Moreover, we prove that a CPWL function is extremal if and only if its Hessian is minimally supported. For the converse, we prove that the density result (which we have only proven for dimension $d = 2$) implies that the closure of the CPWL extreme points contains all extremal points.

math.FA↗

arXiv2022

Coupled Splines for Sparse Curve Fitting

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 optimal solution to the inverse problem is a closed curve with spline components. We then show how to efficiently solve the task using B-splines as basis functions. We extend our problem formulation to curves made of two distinct components with complementary smoothness properties and solve it using hybrid splines. We illustrate the performance of our model on contours of different smoothness. Our experimental results show that we can faithfully reconstruct any general contour using few parameters, even in the presence of imprecisions in the measurements.

eess.IV↗

arXiv2021

Sparsest Univariate Learning Models Under Lipschitz Constraint

Beside the minimization of 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. In our first approach, we use the Lipschitz constant as a regularizer, which results in an implicit tuning of the overall robustness of the learned mapping. In our second approach, we control the Lipschitz constant explicitly using a user-defined upper-bound and make use of a sparsity-promoting regularizer to favor simpler (and, hence, more interpretable) solutions. The theoretical study of the latter formulation is motivated in part by its equivalence, which we prove, with the training of a Lipschitz-constrained two-layer univariate neural network with rectified linear unit (ReLU) activations and weight decay. By proving representer theorems, we show that both problems admit global minimizers that are continuous and piecewise-linear (CPWL) functions. Moreover, we propose efficient algorithms that find the sparsest solution of each problem: the CPWL mapping with the least number of linear regions. Finally, we illustrate numerically the outcome of our formulations.

cs.LG↗

arXiv2021

Measuring Complexity of Learning Schemes Using Hessian-Schatten Total Variation

In this paper, we introduce the Hessian-Schatten total variation (HTV) -- a novel seminorm that quantifies the total "rugosity" of multivariate functions. Our motivation for defining HTV is to assess the complexity of supervised-learning schemes. We start by specifying the adequate matrix-valued Banach spaces that are equipped with suitable classes of mixed norms. We then show that the HTV is invariant to rotations, scalings, and translations. Additionally, its minimum value is achieved for linear mappings, which supports the common intuition that linear regression is the least complex learning model. Next, we present closed-form expressions of the HTV for two general classes of functions. The first one is the class of Sobolev functions with a certain degree of regularity, for which we show that the HTV coincides with the Hessian-Schatten seminorm that is sometimes used as a regularizer for image reconstruction. The second one is the class of continuous and piecewise-linear (CPWL) functions. In this case, we show that the HTV reflects the total change in slopes between linear regions that have a common facet. Hence, it can be viewed as a convex relaxation (l1-type) of the number of linear regions (l0-type) of CPWL mappings. Finally, we illustrate the use of our proposed seminorm.

cs.LG↗

arXiv2021

Convex Optimization in Sums of Banach Spaces

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 components, where each component belongs to its own prescribed Banach space; moreover, the problem is regularized by penalizing some composite norm of the solution. We establish general conditions for existence and derive the generic parametric representation of the solution components. These representations fall into three categories depending on the underlying regularization norm: (i) a linear expansion in terms of predefined "kernels" when the component space is a reproducing kernel Hilbert space (RKHS), (ii) a non-linear (duality) mapping of a linear combination of measurement functionals when the component Banach space is strictly convex, and, (iii) an adaptive expansion in terms of a small number of atoms within a larger dictionary when the component Banach space is not strictly convex. Our approach generalizes and unifies a number of multi-kernel (RKHS) and sparse-dictionary learning techniques for compressed sensing available in the literature. It also yields the natural extension of the classical spline-fitting techniques in (semi-)RKHS to the abstract Banach-space setting.

math.OC↗

arXiv2021

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

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 continuous-domain regularized inverse problem with multiple penalties. We prove that these penalties induce reconstructed signals that indeed take the desired form of the sum of a sparse and a smooth component. We then discretize this problem using Riesz bases, which yields a discrete problem that can be solved by standard algorithms. Our discretization is exact in the sense that we are solving the continuous-domain problem over the search space specified by our bases without any discretization error. We propose a complete algorithmic pipeline and demonstrate its feasibility on simulated data.

eess.SP↗

arXiv2020

Shortest-support Multi-Spline Bases for Generalized Sampling

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-invariant space that can reproduce polynomials up to a given degree $M$. While this property allows for an approximation power of order $(M+1)$, it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least $(M+1)$. Following this result, we introduce the notion of shortest basis of degree $M$, which is motivated by our desire to minimize the computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications such as fast derivative sampling with arbitrarily high approximation power.

math.NA↗

arXiv2020

Duality Mapping for Schatten Matrix Norms

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 H\"older inequality for Schatten norms. We prove in our main result that, for $p\in (1,\infty)$, the duality mapping over the space of real-valued matrices with Schatten-$p$ norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case $p = 1$, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable single-valued function for which we also provide a closed-form expression.

math.FA↗

arXiv2020

Generating Sparse Stochastic Processes Using Matched Splines

We provide an algorithm to generate trajectories of sparse stochastic processes that are solutions of linear ordinary differential equations driven by L\'evy white noises. A recent paper showed that these processes are limits in law of generalized compound-Poisson processes. Based on this result, we derive an off-the-grid algorithm that generates arbitrarily close approximations of the target process. Our method relies on a B-spline representation of generalized compound-Poisson processes. We illustrate numerically the validity of our approach.

math.ST↗

arXiv2020

Wavelet Compressibility of Compound Poisson Processes

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 nonzero wavelet coefficients up to a given scale, what we call the greedy approximation, we exploit the extreme sparsity of the wavelet expansion that derives from the piecewise-constant nature of compound Poisson processes. More precisely, we provide lower and upper bounds for the mean squared error of greedy approximation of compound Poisson processes. We are then able to deduce that the greedy approximation error has a sub-exponential and super-polynomial asymptotic behavior. Finally, we provide numerical experiments to highlight the remarkable ability of wavelet-based dictionaries in achieving highly compressible approximations of compound Poisson processes.

cs.IT↗

arXiv2020

Deep Neural Networks with Trainable Activations and Controlled Lipschitz Constant

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, we first establish a global bound for the Lipschitz constant of neural networks. Based on the obtained bound, we then formulate a variational problem for learning activation functions. Our variational problem is infinite-dimensional and is not computationally tractable. However, we prove that there always exists a solution that has continuous and piecewise-linear (linear-spline) activations. This reduces the original problem to a finite-dimensional minimization where an l1 penalty on the parameters of the activations favors the learning of sparse nonlinearities. We numerically compare our scheme with standard ReLU network and its variations, PReLU and LeakyReLU and we empirically demonstrate the practical aspects of our framework.

cs.LG↗

arXiv2019

Support and Approximation Properties of Hermite Splines

In this paper, we formally investigate two mathematical aspects of Hermite splines which translate to features that are relevant to their practical applications. We first demonstrate that Hermite splines are maximally localized in the sense that their support sizes are minimal among pairs of functions with identical reproduction properties. Then, we precisely quantify the approximation power of Hermite splines for reconstructing functions and their derivatives, and show that they are asymptotically identical to cubic B-splines for these tasks. Hermite splines therefore combine optimal localization and excellent approximation power, while retaining interpolation properties and closed-form expression, in contrast to existing similar approaches. These findings shed a new light on the convenience of Hermite splines for use in computer graphics and geometrical design.

math.NA↗

arXiv2018

Multi-Kernel Regression with Sparsity Constraint

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 learning in a continuous-domain hybrid search space with gTV regularization. We show that the solution admits a multi-kernel expansion with adaptive positions. In this representation, the number of active kernels is upper-bounded by the number of data points while the gTV regularization imposes an $\ell_1$ penalty on the kernel coefficients. Finally, we illustrate numerically the outcome of our theory.

cs.LG↗

arXiv2018

Wavelet Analysis of the Besov Regularity of L\'evy White Noises

We characterize the local smoothness and the asymptotic growth rate of the L\'evy 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 smoothness via the Blumenthal-Getoor indices of the L\'evy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-alpha-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the L\'evy white noise. Previous analyses only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires us to determine in which Besov spaces a given L\'evy white noise is (almost surely) not. Our methods are based on the wavelet-domain characterization of Besov spaces and precise moment estimates for the wavelet coefficients of the L\'evy white noise.

math.PR↗