arXiv subjects

Alexander Kolpakov

Publications and source records attributed to Alexander Kolpakov.

At least 19 recordsLinked to original sources

arXiv2026

Random Knots via Stiefel manifolds

A fixed simplex, randomly projected into three dimensions and joined in Hamiltonian order, produces a rich and unusually tractable model of random stick knots. We prove that Gaussian projections and Haar-random Stiefel projections have exactly the same knot-type law, despite having different metric shapes, and that at every stick budget the model gives positive probability to precisely the knot types realizable with that many sticks. Its linear-algebraic structure yields an exact marginal distance law, an exact mean planar-crossing count, and crossing concentration, while in the first nontrivial six-stick case the complete tetrahedral sign pattern gives an exact unknot-versus-handed-trefoil classifier for every generic sample. The result is a direct bridge from random projections and finite sign geometry to the topology of random knots.

math.GT↗

arXiv2026

Discriminant Varieties for Stick Knots and Links

How many knot types can be built from a fixed budget of straight sticks? We prove that the answer has factorial-scale growth, settling its order for the first time. No previously published general upper bound improves on the exponential-in-the-square estimate obtained from crossing-number enumeration; we replace it with a factorial-scale upper bound, which is optimal at the level of growth order. The proof turns polygonal self-intersection into a sparse real-algebraic chamber problem in only linearly many dimensions, while a complementary braid construction supplies factorially many distinct knots. The result creates a direct bridge between knot topology, real algebraic geometry, fewnomial structure, and permutation combinatorics.

math.GT↗

arXiv2026

DiRe-RAPIDS: Topology-faithful dimensionality reduction at scale

Dimensionality reduction methods such as UMAP and t-SNE are central tools for visualising high-dimensional data, but their local-neighborhood objectives can preserve sampling noise while distorting global topology. We show that standard local metrics reward this noise memorisation: top-performing embeddings invent cycles and disconnected islands absent from the data. We introduce a topology-faithfulness benchmark based on noisy manifolds with known homology, tune DiRe against it, and find Pareto-optimal configurations that match or beat GPU-accelerated UMAP on classification while recovering exact first Betti numbers on stress tests. On 723K arXiv paper embeddings, DiRe preserves 3-4 times more topological structure than UMAP at comparable wall-clock.

cs.LG↗

arXiv2025

Burau representation, Squier's form, and non-Abelian anyons

We introduce a frequency-tunable, two-dimensional non-Abelian control of operation order constructed from the reduced Burau representation of the braid group $B_3$, specialised at $t=e^{i\omega}$ and unitarized by Squier's Hermitian form. Coupled to two non-commuting qubit unitaries $A$, $B$, the resulting switch admits a closed expression for the single-shot Helstrom success probability and a fixed-order ceiling $p_{\mathrm{fixed}}$, defining the fixed-order ceiling $p_{\mathrm{fixed}}^*$ and the witness gaps $\Delta_{\rm sw}(\omega)=p_{\mathrm{switch}}(\omega)-p_{\mathrm{fixed}}^*$ and $\Delta_{\rm test}(\omega)=p_{\mathrm{test}}(\omega)-p_{\mathrm{fixed}}^*$. The non-Abelian mixers can either enhance or suppress the bare switch advantage, which we quantify by the interference contrast $\Delta_{\rm int}(\omega):=\Delta_{\rm test}(\omega)-\Delta_{\rm sw}(\omega)=p_{\rm test}(\omega)-p_{\rm switch}(\omega)$. Across the Squier positivity region, $\Delta_{\rm int}(\omega)$ takes both positive (constructive) and negative (destructive) values, a hallmark of matrix-valued (non-Abelian) order control, while $\Delta_{\rm sw}(\omega)>0$ certifies algebraic causal non-separability. Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal $B_3$ braid control that reproduces the characteristic interference pattern expected from a \emph{Gedankenexperiment} in anyonic statistics.

quant-ph↗

arXiv2025

Multiplicative Turing Ensembles, Pareto's Law, and Creativity

We study integer-valued multiplicative dynamics driven by i.i.d. prime multipliers and connect their macroscopic statistics to universal codelengths. We introduce the Multiplicative Turing Ensemble (MTE) and show how it arises naturally -- though not uniquely -- from ensembles of probabilistic Turing machines. Our modeling principle is variational: taking Elias' Omega codelength as an energy and imposing maximum entropy constraints yields a canonical Gibbs prior on integers and, by restriction, on primes. Under mild tail assumptions, this prior induces exponential tails for log-multipliers (up to slowly varying corrections), which in turn generate Pareto-type tails for additive gaps, with the survival exponent shifted by summation over primes. We also prove time-average laws for the Omega codelength along MTE trajectories. Empirically, Debian, PyPI, and CRAN package-size histograms have fitted Omega slopes well below the pure-Omega value $\log 2$, indicating heavier-than-pure-Omega tails within this energy scale. Taken together, the theory--data comparison suggests a qualitative split: machine-adapted regimes (Gibbs-aligned, finite first moment) exhibit clean averaging behavior, whereas human-generated complexity appears to sit beyond this regime, with tails heavy enough to produce an unbounded first moment, and therefore no averaging of the same kind.

cs.IT↗

arXiv2025

Loss-Complexity Landscape and Model Structure Functions

We develop a framework for dualizing the Kolmogorov structure function $h_x(\alpha)$, which then allows using computable complexity proxies. We establish a mathematical analogy between information-theoretic constructs and statistical mechanics, introducing a suitable partition function and free energy functional. We explicitly prove the Legendre-Fenchel duality between the structure function and free energy, showing detailed balance of the Metropolis kernel, and interpret acceptance probabilities as information-theoretic scattering amplitudes. A susceptibility-like variance of model complexity is shown to peak precisely at loss-complexity trade-offs interpreted as phase transitions. Practical experiments with linear and tree-based regression models verify these theoretical predictions, explicitly demonstrating the interplay between the model complexity, generalization, and overfitting threshold.

cs.IT↗

arXiv2025

Lagrangians, Renormalization, and Quantization in Prefix Coding

We develop a statistical mechanics framework for prefix coding based on variational principles, renormalization, and quantization. A Lagrangian formulation of entropy-optimal encoding under the Kraft-McMillan constraint yields a Gibbs-type implied distribution and completeness of the optimal code. A renormalization operator acting on codeword distribution laws produces a coarse-graining flow whose fixed points have iterated-log structure; discrete quantizations of these fixed points include Elias' $\omega$ code as a special case. Extending the theory to mixed discrete-continuous source laws, we show how continuous codelength functions can be quantized into countable prefix codes and derive resolution-adjusted entropy bounds together with Heisenberg-type and Boltzmann-type relations. This provides a unified and physically motivated view of universal coding, with Elias' $\omega$ code as a guiding example.

cs.IT↗

arXiv2025

Fast Geometric Embedding for Node Influence Maximization

Computing classical centrality measures such as betweenness and closeness is computationally expensive on large-scale graphs. In this work, we introduce an efficient force layout algorithm that embeds a graph into a low-dimensional space, where the radial distance from the origin serves as a proxy for various centrality measures. We evaluate our method on multiple graph families and demonstrate strong correlations with degree, PageRank, and paths-based centralities. As an application, it turns out that the proposed embedding allows one to find high-influence nodes in a network, and provides a fast and scalable alternative to the standard greedy algorithm.

cs.SI↗

arXiv2025

Dimensionality reduction for homological stability and global structure preservation

We propose DiRe, a force-directed dimensionality reduction framework designed to preserve global structure and homological features while remaining practical on modern hardware. The method combines an initial embedding with a graph-based layout optimization and evaluates the resulting low-dimensional representation using local distortion, context preservation, and persistent homology measures. Across the benchmark suite considered here, DiRe provides a complementary tradeoff to UMAP and tSNE: it is designed less as a purely local visualization heuristic and more as a framework for embeddings whose large-scale geometry can be quantified through Betti curves and persistence diagrams.

cs.LG↗

arXiv2025

Benford's Law from Turing Ensembles and Integer Partitions

We develop two complementary generative mechanisms that explain when and why Benford's first-digit law arises. First, a probabilistic Turing machine (PTM) ensemble induces a geometric law for codelength. Maximizing its entropy under a constraint on halting length yields Benford statistics. This model shows a phase transition with respect to the halt probability. Second, a constrained partition model (Einstein-solid combinatorics) recovers the same logarithmic profile as the maximum-entropy solution under a coarse-grained entropy-rate constraint, clarifying the role of non-ergodicity (ensemble vs. trajectory averages). We also perform numerical experiments that corroborate our conclusions.

cs.IT↗

arXiv2025

A ripple in time: a discontinuity in American history

In this technical note we suggest a novel approach to discover temporal (related and unrelated to language dilation) and personality (authorship attribution) aspects in historical datasets. We exemplify our approach on the State of the Union addresses given by the past 42 US presidents: this dataset is known for its relatively small amount of data, and high variability of the size and style of texts. Nevertheless, we manage to achieve about 95\% accuracy on the authorship attribution task, and pin down the date of writing to a single presidential term.

cs.CL↗

arXiv2024

The Information Geometry of UMAP

In this note we highlight some connections of UMAP to the basic principles of Information Geometry. Originally, UMAP was derived from Category Theory observations. However, we posit that it also has a natural geometric interpretation.

cs.CG↗

arXiv2024

On the impossibility of discovering a formula for primes using AI

The present work explores the theoretical limits of Machine Learning (ML) within the framework of Kolmogorov's theory of Algorithmic Probability, which clarifies the notion of entropy as Expected Kolmogorov Complexity and formalizes other fundamental concepts such as Occam's razor via Levin's Universal Distribution. As a fundamental application, we develop Maximum Entropy methods that allow us to derive the Erdős-Kac Law and Hardy-Ramanujan theorem in Probabilistic Number Theory, and establish the impossibility of discovering a formula for primes using Machine Learning via the Prime Coding Theorem.

cs.CC↗

arXiv2024

Robust affine point matching via quadratic assignment on Grassmannians

Robust Affine Matching with Grassmannians (RoAM) is a new algorithm to perform affine registration of point clouds. The algorithm is based on minimizing the Frobenius distance between two elements of the Grassmannian. For this purpose, an indefinite relaxation of the Quadratic Assignment Problem (QAP) is used, and several approaches to affine feature matching are studied and compared. Experiments demonstrate that RoAM is more robust to noise and point discrepancy than previous methods.

cs.CV↗

arXiv2024

Kleinian sphere packings, reflection groups, and arithmeticity

In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study "properly integral" packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher-dimensional hyperbolic polyhedra.

math.GT↗

arXiv2024

Thin hyperbolic reflection groups

We study a family of Zariski dense finitely generated discrete subgroups of $\mathrm{Isom}(\mathbb{H}^d)$, $d \geqslant 2$, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an application we obtain a general description of all thin hyperbolic reflection groups. In particular, we show that the Vinberg algorithm applied to a non-reflective Lorentzian lattice gives rise to an infinite sequence of thin reflection subgroups in $\mathrm{Isom}(\mathbb{H}^d)$, for any $d \geqslant 2$. Moreover, every such group is a subgroup of a group produced by the Vinberg algorithm applied to a Lorentzian lattice independently on the latter being reflective. As a consequence, all thin hyperbolic reflection groups are enumerable.

math.GR↗

arXiv2023

An approach to robust ICP initialization

In this note, we propose an approach to initialize the Iterative Closest Point (ICP) algorithm to match unlabelled point clouds related by rigid transformations. The method is based on matching the ellipsoids defined by the points' covariance matrices and then testing the various principal half-axes matchings that differ by elements of a finite reflection group. We derive bounds on the robustness of our approach to noise and numerical experiments confirm our theoretical findings.

cs.CV↗

arXiv2023

Expansion properties of Whitehead moves on cubic graphs

The present note concerns the "graph of graphs" that has cubic graphs as vertices connected by edges represented by the so-called Whitehead moves. Here, we prove that the outer-conductance of the graph of graphs tends to zero as the number of vertices tends to infinity. This answers a question of K. Rafi in the negative.

math.CO↗