Computation and Applications of Euclidean and Normed Representations of Massive Data

Max Ovsiankin

Home/Archive/arXiv

arXiv · 2609.06329

Computation and Applications of Euclidean and Normed Representations of Massive Data

Max Ovsiankin

View original source ↗Read source document ↗Export metadata ↓

Abstract

This thesis investigates Euclidean-space and $\ell_p$-norm representations of different forms of data, with a focus on efficient algorithms for computing these representations in settings where the amount of data is very large. The applications of such representations are also discussed: they may be used to summarize the data in a more compact form for downstream tasks while preserving its most salient properties; and further, they may also be used to extract insights about the data that may not be apparent in its original form. This thesis presents novel algorithms for computing Euclidean representations in the case where the data comes with linear structure, as well as in the case where the data comes only with metric, or distance structure. In addition, we give algorithms for computing $\ell_p$-norm representations in linear structured cases. The analyses of these algorithms use tools from geometry, probability, and optimization, and these tools are used to illuminate other algorithms for similar problems.

Explore related subjects

cs.DS math.OC

Keep this discovery

BibTeX RIS

Max Ovsiankin. 2026-09-06. Computation and Applications of Euclidean and Normed Representations of Massive Data. https://arxiv.org/abs/2609.06329

Cite the original work for its findings. Save a collection to share your selection of sources.

Discover connections

cs.DS subject math.OC subject

Connections use source metadata and explicit phrase matches, not verified experimental comparisons.

KEEP EXPLORING

Related papers

arXiv2026

Nonlinear Dynamics In Optimization Landscape of Shallow Neural Networks with Tunable Leaky ReLU

In this work, we study the nonlinear dynamics of a shallow neural network trained with mean-squared loss and leaky ReLU activation. Under Gaussian inputs and equal layer width k, (1) we establish, based on the equivariant gradient degree, a theoretical framework, applicable to any number of neurons k>= 4, to detect bifurcation of critical points with associated symmetries from global minimum as leaky parameter $α$ varies. Typically, our analysis reveals that a multi-mode degeneracy consistently occurs at the critical number 0, independent of k. (2) As a by-product, we further show that such bifurcations are width-independent, arise only for nonnegative $α$ and that the global minimum undergoes no further symmetry-breaking instability throughout the engineering regime $α$ in range (0,1). An explicit example with k=5 is presented to illustrate the framework and exhibit the resulting bifurcation together with their symmetries.

math.OC↗

arXiv2025

Automated algorithm design for convex optimization problems with linear equality constraints

Synthesis of optimization algorithms typically follows a {\em design-then-analyze\/} approach, which can obscure fundamental performance limits and hinder the systematic development of algorithms that operate near these limits. Recently, a framework grounded in robust control theory has emerged as a powerful tool for automating algorithm synthesis. By integrating design and analysis stages, fundamental performance bounds are revealed and synthesis of algorithms that achieve them is enabled. In this paper, we apply this framework to design algorithms for solving strongly convex optimization problems with linear equality constraints. Our approach yields a single-loop, gradient-based algorithm whose convergence rate is independent of the condition number of the constraint matrix. This improves upon the best known rate within the same algorithm class, which depends on the product of the condition numbers of the objective function and the constraint matrix.

math.OC↗

arXiv2026

On two proofs of $d^2$ mixing of weighted Dikin walks

We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, $\barν$-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an $\widetilde O(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\widetilde O(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger $χ^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an $\widetilde O(d^2)$ mixing bound in $χ^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.

cs.DS↗