Related papers: On the adaptive spectral approximation of function…
For planar and cubic Ising models, we examined two ways of approximation of a spectral density that describes a degeneracy of energy levels. We approximated the exponent of the spectral density by polynomials of even degrees and using our…
In this work, an efficient approximation scheme has been proposed for getting accurate approximate solution of nonlinear partial differential equations with constant or variable coefficients satisfying initial conditions in a series of…
Smoothing (and decay) spacetime estimates are discussed for evolution groups of self-adjoint operators in an abstract setting. The basic assumption is the existence (and weak continuity) of the spectral density in a functional setting.…
Computing eigenvalues of very large matrices is a critical task in many machine learning applications, including the evaluation of log-determinants, the trace of matrix functions, and other important metrics. As datasets continue to grow in…
The proximal gradient algorithm for minimizing the sum of a smooth and a nonsmooth convex function often converges linearly even without strong convexity. One common reason is that a multiple of the step length at each iteration may…
Spectral dimensionality reduction algorithms are widely used in numerous domains, including for recognition, segmentation, tracking and visualization. However, despite their popularity, these algorithms suffer from a major limitation known…
We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal…
The convergence of stochastic gradient descent is highly dependent on the step-size, especially on non-convex problems such as neural network training. Step decay step-size schedules (constant and then cut) are widely used in practice…
Convolutional autoencoders have emerged as popular methods for unsupervised defect segmentation on image data. Most commonly, this task is performed by thresholding a pixel-wise reconstruction error based on an $\ell^p$ distance. This…
We study the problem of approximating the level set of an unknown function by sequentially querying its values. We introduce a family of algorithms called Bisect and Approximate through which we reduce the level set approximation problem to…
Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation…
In distributed optimization problems, a technique called gradient coding, which involves replicating data points, has been used to mitigate the effect of straggling machines. Recent work has studied approximate gradient coding, which…
Bilevel optimization is a central tool in machine learning for high-dimensional hyperparameter tuning. Its applications are vast; for instance, in imaging it can be used for learning data-adaptive regularizers and optimizing forward…
Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness…
We consider the problem of decomposing a piecewise constant function on the circle into a sum of indicator functions of closed circular disks in the plane, whose number and location are not a priori known. This represents a situation where…
Learning compatible representations enables the interchangeable use of semantic features as models are updated over time. This is particularly relevant in search and retrieval systems where it is crucial to avoid reprocessing of the gallery…
We explore conditions for when the gradient of a deep declarative node can be approximated by ignoring constraint terms and still result in a descent direction for the global loss function. This has important practical application when…
This investigation seeks to establish the practicality of numerical frame approximations. Specifically, it develops a new method to approximate the inverse frame operator and analyzes its convergence properties. It is established that…
In functional linear regression, the slope ``parameter'' is a function. Therefore, in a nonparametric context, it is determined by an infinite number of unknowns. Its estimation involves solving an ill-posed problem and has points of…
We demonstrate that accurate computation of the spectral proper orthogonal decomposition (SPOD) critically depends on the choice of frequency resolution. Using both artificially generated data and large-eddy simulation data of a turbulent…