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We consider various notions of completeness in symplectic topology and ask two related questions. Does a complete open symplectic manifold remain complete after excising a subset? Can two sets be made arbitrarily far apart by adjusting the…
Linear encoding of sparse vectors is widely popular, but is commonly data-independent -- missing any possible extra (but a priori unknown) structure beyond sparsity. In this paper we present a new method to learn linear encoders that adapt…
Several characterizations of complex ellipsoids among convex bodies in Cn, in terms of their sections and projections are proved. Characterizing complex symmetry in similar terms is an important tool.
We consider the problem of sampling from the posterior distribution of a $d$-dimensional coefficient vector $\boldsymbol{\theta}$, given linear observations $\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\theta}+\boldsymbol{\varepsilon}$. In…
The task of this survey is to present various results on intersection patterns of convex sets. One of main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so called…
We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the…
Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary $D$. This problem is now understood to be well-posed and…
We study the problem of reconstructing a convex body using only a finite number of measurements of outer normal vectors. More precisely, we suppose that the normal vectors are measured at independent random locations uniformly distributed…
Suppose a given observation matrix can be decomposed as the sum of a low-rank matrix and a sparse matrix (outliers), and the goal is to recover these individual components from the observed sum. Such additive decompositions have…
Random projections offer an appealing and flexible approach to a wide range of large-scale statistical problems. They are particularly useful in high-dimensional settings, where we have many covariates recorded for each observation. In…
Introducing sets of constraints, we define new classes of random-matrix ensembles, the constrained Gaussian unitary (CGUE) and the deformed Gaussian unitary (DGUE) ensembles. The latter interpolate between the GUE and the CGUE. We derive a…
We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a convex subset $\mathcal{C}$ of $\mathbb{R}^d$. We adopt a minimax point of view and our primary…
Learning with few samples is a major challenge for parameter-rich models like deep networks. In contrast, people learn complex new concepts even from very few examples, suggesting that the sample complexity of learning can often be reduced.…
High-dimensional sparse data present computational and statistical challenges for supervised learning. We propose compact linear sketches for reducing the dimensionality of the input, followed by a single layer neural network. We show that…
This contribution proposes a recursive set-membership method for the ellipsoidal state characterization for discrete-time linear time-varying models with additive unknown disturbances vectors, bounded by possibly degenerate zonotopes and…
We study the correlation between the locations of galaxy-galaxy strong lensing candidates and tracers of large-scale structure from both weak lensing or X-ray emission. The COSMOS survey is a unique data set, combining deep, high resolution…
Compressive sensing is a powerful technique for recovering sparse solutions of underdetermined linear systems, which is often encountered in uncertainty quantification analysis of expensive and high-dimensional physical models. We perform…
In this work, we address the unsupervised classification issue by exploiting the general idea of Random Projection Ensemble. Specifically, we propose to generate a set of low dimensional independent random projections and to perform…
Because hyperspectral remote sensing images contain a lot of redundant information and the data structure is highly non-linear, leading to low classification accuracy of traditional machine learning methods. The latest research shows that…
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for…