Abstract: Consider a real vector space V and a finite group G acting unitary on V. We study the general problem of constructing a stable embedding, whose domain is the quotient of the vector space modulo the group action, and whose target space is an Euclidean space. The embedding scheme we propose is based on taking a fixed subset out of sorted coorbit ()_g , where w_i are appropriate vectors. Finally, we show that injectivity on quotient space implies stability.
Abstract: In this talk, we will introduce the HRT conjecture and prove it for two simple cases. Then, we will introduce the Fock space of entire functions and use it to show that the HRT conjecture holds for point configurations where all but one point lie on a line. This talk is based on a book chapter by Daniel W. Stroock.
Abstract: This talk is a continuation of a presentation with the same name last week. By viewing the HRT conjecture from the point of view of the Fock space, we show that it holds for a dense subset of the square-integrable signals. Thereafter, we present a characterisation of this dense subset. This talk is based on a book chapter by Daniel W.~Stroock.
Abstract: This talk is a continuation of a presentation with the same name last week. By viewing the HRT conjecture from the point of view of the Fock space, we show that it holds for a dense subset of the square-integrable signals. Thereafter, we present a characterization of this dense subset. This talk is based on a book chapter by Daniel W. Stroock.
Abstract: This talk is based on the paper by Bownick and Speegle (2012) with the same title. We establish the linear independence of time-frequency translates of functions with faster than exponential decay, under some additional restrictions
Abstract: This talk is based on the paper by Bownick and Speegle (2012) with the same title. We establish the linear independence of time-frequency translates of functions with faster than exponential decay, under some additional restrictions.
Abstract: This talk is based on the paper by Bownick and Speegle (2012) with the same title. We establish the linear independence of time-frequency translates of functions with faster than exponential decay, under some additional restrictions.
Abstract: Many inverse problems in science and engineering boil down to the solution of variationally regularized optimization problems containing a fidelity term measuring the fit to the data weighted against a regularization term penalizing the complexity of the solution. In this context, it is important to choose a good hyperparameter for weighting of the fidelity against the regularization to obtain stable and accurate solutions. In this talk, I will study this hyperparameter choice problem through the lens of bilevel optimization, an optimization framework where the constraint is also an optimization problem. In particular, I will present the results of Holler et. al (2018) and Ehrhardt et. al (2023) on: (1) existence of solutions to the bilevel problem and (2) positivity of solutions in the single weighting term case. Time-permitting, I will present some results on the fast computation of solutions using first-order methods.
Abstract: We will prove a stability estimate for the hyperparameter tuning bilevel problem stated in the first talk. As a corollary, the existence of solutions to the upper level problem will follow. Time permitting we will also discuss the positivity of solutions to the bilevel problem, with a focus on the denoising case.
Abstract: A problem posed by H. Feichtinger (and subsequently modified by C. Heil and D. Larson) asks whether a type of positive-definite integral operators with $M_1$ kernel admits a rank-one decomposition series that is also strongly square-summable in $M_1$. In this first talk, we will approach this problem by considering its matrix (and finite-dimensional) variant and analyzing several functionals that measure the optimality of such decomposition. Some of the results are based on the joint work with Radu Balan.
Abstract: In this talk we present a CUR matrix approximation that uses a novel convex optimization formulation to select the columns and rows of the data matrix for inclusion in C and R, respectively. We discuss implementation of the algorithm using the surrogate functional of Daubechies et al. [Communications on Pure and Applied Mathematics, 57.11 (2004)] and extend the theoretical guarantees of this approach to our formulation. Applications using CUR as a feature selection method for classification will be shown, if time. In addition, the proximal operator of the L-infinity norm is used in our CUR algorithm. We present a neural network approximation to this proximal operator that uses a novel feature selection process based on moments of the input data in order to allow vectors of varying lengths to be input into the network.
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