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Leonid Bedratyuk

Publications and source records attributed to Leonid Bedratyuk.

At least 19 recordsLinked to original sources
arXiv2026

Polynomial joint first-order differential projective invariants

We study polynomial absolute and relative joint first-order differential projective invariants for configurations of $n$ points in the plane. The main approach is based on passing to a homogeneous vector--covector representation, which reduces the problem to the classical invariant theory of the group $SL(3,\mathbb C)$. The algebra of polynomial absolute invariants is described and shown to be generated by cyclic invariants. For polynomial relative invariants of weight $-1$, a graded description is obtained, their finite generation as a module over the algebra of absolute invariants is proved, and an explicit finite generating system is constructed. For $n=3$, the corresponding module is shown to be free of rank $1$. For $n=4$, a minimal homogeneous generating system consisting of $39$ elements is constructed. The obtained results complement the rational theory of joint projective differential invariants by its polynomial counterpart and provide an algebraic foundation for the further construction of projectively invariant integral characteristics.

math.RA↗
arXiv2026

The Action of the Lie Algebra $\mathfrak{sl}_n$ on Colored Graphs and Multicolored Johnson Graphs

We consider the space of $(n-1)$-colored graphs on a fixed set of $N$ vertices. Each edge position of the complete graph $K_N$ has $n$ possible states: the absence of an edge and $n-1$ colors. This gives a natural identification of the space of such graphs with the tensor power $(\mathbb C^n)^{\otimes m}$, where $m=\binom N2$, and defines on it the diagonal action of the Lie algebra $\mathfrak{gl}_n$, and, after restriction, the action of $\mathfrak{sl}_n$. For a fixed profile $α=(α_0,\dots,α_{n-1})$, we consider the graph $J(m;α)$ whose vertices are colored graphs of this profile and whose adjacency is defined by a single exchange of states in two edge positions. This graph is the transposition graph on the set of words with fixed profile, also known as the \emph{multislice}. The main result is an expression of the adjacency operator in terms of the root operators of $\mathfrak{sl}_n$ and a derivation of its spectrum by means of the quadratic Casimir operator of $\mathfrak{gl}_n$ and the Schur--Weyl decomposition. It is proved that the adjacency operator belongs to the center of the algebra $\End_{S_m}(\mathcal C_α)$. The contribution of each spectral block to the multiplicity of the corresponding eigenvalue is described in terms of a Kostka number and the dimension of a Specht module. For $n=2$, one obtains the classical Johnson graph and its known spectrum. As applications, a formula for the valency is established, connectivity is proved, the Hoffman bound for independent sets is obtained, and the three-state case is considered in detail; in this case the natural symmetrized subspace realizes the module $\Sym^m(\mathbb C^3)$.

math.CO↗
arXiv2026

The Graph Algebra I: Representation-Theoretic Structure

The paper studies the graph algebra whose monomial basis is naturally indexed by simple graphs on a fixed set of vertices. This algebra is at the same time the algebra of pseudo-Boolean functions on the Boolean cube and a natural object of algebraic combinatorics, related to the Boolean lattice of subsets of the edge set of the complete graph. The main aim of the paper is to study two compatible representation-theoretic structures on this algebra: the action of the Lie algebra $\mathfrak{sl}_2$, arising from the operators of adding and deleting one edge, and the action of the pair group $S_n^{(2)}$, induced by the renumbering of vertices. It is proved that the graph algebra with this $\mathfrak{sl}_2$-action is isomorphic to a tensor power of the standard two-dimensional $\mathfrak{sl}_2$-module, and on this basis its decomposition into irreducible $\mathfrak{sl}_2$-modules is obtained. Primitive spaces, that is, the kernels of the edge-deletion operator on rank components, are also described, and it is shown that they have a natural interpretation in terms of two-row Specht modules. It is then established that the $\mathfrak{sl}_2$-action commutes with the action of the pair group. It follows that the space of graph invariants also inherits the structure of an $\mathfrak{sl}_2$-module. Using Schur--Weyl duality, primitive invariants are described through the fixed parts of the restrictions of two-row Specht modules from the full symmetric group on the edge set to the pair group. As a consequence, the classical enumeration of non-isomorphic graphs by the number of edges receives a representation-theoretic refinement: the orbital components entering the Burnside--Polya formula decompose into natural primitive contributions associated with the $\mathfrak{sl}_2$-structure and two-row Specht modules.

math.CO↗
arXiv2026

MSIQ: Moment-based Scale-Invariant Quality Measure for Single Image Super-Resolution

Assessing the quality of single image super-resolution (SISR) results remains an open methodological problem. Common full-reference metrics (PSNR, SSIM, LPIPS) do not explicitly evaluate the preservation of the geometric structure of images, which is critical for the correctness of scale-based reconstruction. In addition, they require the forced alignment of images to the same size (\textit{forced resizing}), which introduces an external interpolation error into the evaluation process. This paper proposes a diagnostic scale-invariant quality measure, MSIQ (\textit{Moment-based Scale-Invariant Quality}), based on the comparison of normalized central geometric moments of two images. MSIQ enables direct comparison of images with different spatial resolutions without resizing, is mathematically deterministic (\textit{model-free}), and has an analytical form. To provide a theoretical basis for the approach, we introduce a conceptual distinction between the ability of metrics to monotonically track degradation (\textit{tracking ability}) and their geometric selectivity (\textit{geometric specificity}). The experimental validation confirmed the stability of MSIQ under uniform scaling and, at the same time, revealed the high sensitivity of traditional metrics to the choice of interpolation method. The results show that MSIQ has pronounced geometric selectivity: the proposed measure effectively separates geometric deformations from non-geometric artifacts, in particular JPEG compression, unlike pixel-based and perceptual metrics. It is also shown that the response of MSIQ to structural perturbations remains stable across different classes of SR algorithms, including DNN models with different architectures. The proposed measure is a complementary diagnostic tool for domains where geometric fidelity has priority, in particular medical imaging and remote sensing.

cs.CV↗
arXiv2026

Controlled Paraphrase Geometry in Sentence Embedding Space: Local Manifold Modeling and Latent Probing

The paper studies the local geometry of embedding clouds induced by \emph{controlled local classes of semantically close sentences}. The central question is how controlled paraphrase-like semantic variation is organized in sentence embedding space and whether this local structure can be explicitly modeled by low-degree fitted carriers. We introduce a local geometric modeling scheme based on affine, quadratic, and cubic fitted models. We also use a surface-based latent probing procedure that constructs synthetic latent points in a reduced local PCA space with respect to the fitted carrier. The procedure is intended as an offline method for representation-space analysis, local manifold modeling, and geometry-aware latent probing. Generated latent points are evaluated using criteria that measure consistency with the fitted surface, preservation of neighborhood structure, agreement with the empirical distribution, stability of Hessian-based second-order shape descriptors, and stability of fitted-model coefficients. Experiments on controlled sets of semantically close sentences show that nonlinear local models describe embedding clouds more accurately than affine models. Surface-based generation provides strong fitted-geometry fidelity, including surface consistency, Hessian-based shape consistency, and coefficient consistency. Downstream experiments show that geometric validity of synthetic latent points does not automatically translate into improved classification performance. The results support explicit local geometric modeling of sentence embedding space and highlight the need to distinguish geometric validity from discriminative utility. As a resource contribution, we introduce \textbf{CoPaGE-300K}, a controlled template-based dataset of semantically close sentence variants with slot-level annotations and precomputed sentence embeddings.

cs.CL↗
arXiv2026

LAG-XAI: A Lie-Inspired Affine Geometric Framework for Interpretable Paraphrasing in Transformer Latent Spaces

Modern Transformer-based language models achieve strong performance in natural language processing tasks, yet their latent semantic spaces remain largely uninterpretable black boxes. This paper introduces LAG-XAI (Lie Affine Geometry for Explainable AI), a novel geometric framework that models paraphrasing not as discrete word substitutions, but as a structured affine transformation within the embedding space. By conceptualizing paraphrasing as a continuous geometric flow on a semantic manifold, we propose a computationally efficient mean-field approximation, inspired by local Lie group actions. This allows us to decompose paraphrase transitions into geometrically interpretable components: rotation, deformation, and translation. Experiments on the noisy PIT-2015 Twitter corpus, encoded with Sentence-BERT, reveal a "linear transparency" phenomenon. The proposed affine operator achieves an AUC of 0.7713. By normalizing against random chance (AUC 0.5), the model captures approximately 80% of the non-linear baseline's effective classification capacity (AUC 0.8405), offering explicit parametric interpretability in exchange for a marginal drop in absolute accuracy. The model identifies fundamental geometric invariants, including a stable matrix reconfiguration angle (~27.84°) and near-zero deformation, indicating local isometry. Cross-domain generalization is confirmed via direct cross-corpus validation on an independent TURL dataset. Furthermore, the practical utility of LAG-XAI is demonstrated in LLM hallucination detection: using a "cheap geometric check," the model automatically detected 95.3% of factual distortions on the HaluEval dataset by registering deviations beyond the permissible semantic corridor. This approach provides a mathematically grounded, resource-efficient path toward the mechanistic interpretability of Transformers.

cs.CL↗
arXiv2026

Discrete Moving Frames, Semi-Algebraic Invariants and the Graph Canonization Problem

This paper develops an invariant--geometric interpretation of the canonization problem for simple undirected weighted graphs based on the {discrete moving frame method} for finite groups. We consider the action of the {pair group} $S_n^{(2)}$ on the space of edge weights of a graph. It is emphasized that the classical algebraic approach aimed at describing the ring of polynomial invariants of this action quickly becomes computationally impractical due to the explosive growth in the number and degrees of generators. The main result is a formalization of a canonical labeling of a graph as a {discrete moving frame} in the sense of Olver: a discrete orbit cross-section is fixed, in particular by a lexicographic rule, and for each configuration of edge weights one defines a permutation in $S_n^{(2)}$ that maps it to its canonical representative. The coordinates of the canonical representative are interpreted as a {complete system of invariants} for the action of $S_n^{(2)}$ that separates orbits, i.e., isomorphism classes of graphs. It is shown that the invariants obtained via such orbit canonization are of a non-algebraic nature and belong to the class of {semi-algebraic functions}. We do not propose a new computational algorithm; instead, we provide a rigorous theoretical foundation for the very concept of canonization by viewing it as a process of constructing a discrete moving frame and the corresponding system of semi-algebraic invariants.

math.CO↗
arXiv2026

Relative Invariants from Moving Frames on an Extended Manifold

A constructive modification of the moving frame method is developed in this paper for the construction of relative invariants of regular Lie group actions. Let a relative invariant $I$ of weight $ω$ transform according to the rule $$ I(g \cdot \boldsymbol x) = μ(g, \boldsymbol x)^ω I(\boldsymbol x), $$ where $μ: G \times \mathcal{M} \to \mathbb{R}^\times$ is a scalar multiplier (1-cocycle). It is shown that the cocycle property of $μ$ is equivalent to the well-definedness of the twisted group action on the extended manifold $\widehat{\mathcal{M}} = \mathcal{M} \times \mathbb{R}^\times$, and that relative invariants on $\mathcal{M}$ are in one-to-one correspondence with absolute invariants of this action on $\widehat{\mathcal{M}}$. The main result is that, given a moving frame, the invariantization of the multiplier is a canonical relative invariant of weight $-1$. This enables the constructive realization of any weight and yields an explicit formula for an arbitrary relative invariant in terms of the fundamental absolute invariants and the invariantized multiplier. Examples are provided to demonstrate the application of the proposed approach for the projective group $PGL(3, \mathbb{R})$.

math.RA↗
arXiv2025

Joint Projective Invariants on First Jet Spaces of Point Configurations via Moving Frames

We consider the action of the projective group $PGL(3,\mathbb{R})$ on the $n$-fold first-order jet space of point configurations on the plane. Using the method of moving frames, we construct an explicit complete generating set for the field of absolute first-order joint projective differential invariants $\mathcal{I}_{n,0}$ for any $n \ge 3$. This approach provides a unified construction for all $n$, immediately ensuring functional independence of the fundamental invariants and yielding formulas suitable for both symbolic and numerical implementation. Next, we study the field of relative first-order invariants $\mathcal{I}_n$ with Jacobian multiplier. It is shown that the invariantization of the Jacobian under the projective action yields a primitive element of the field extension $\mathcal{I}_n / \mathcal{I}_{n,0}$. Finally, we introduce a multiplicative cochain complex $C^\bullet$ associated with the action of $PGL(3,\mathbb{R})$ on the jet space, and show that the invariantization operator induced by the moving frame generates an explicit contracting homotopy. This provides a constructive proof of the vanishing of higher cohomology and an interpretation of the "defect" of invariantization as an exact cocycle in $C^\bullet$.

math.RA↗
arXiv2025

First order joint differential projective invariants

We present a complete algebraic description of the field of first-order joint projective invariants for configurations of \( n \) points in the plane, under the natural diagonal action of the projective group \( PGL(3,\mathbb{R}) \). For \( n > 1 \), we construct an explicit minimal generating set for the field of absolute invariants and prove its algebraic independence. We further determine the structure of the full field of invariants as a simple algebraic extension of field of absolute invariants, generated by a single primitive relative invariant of weight~$-1$, for which we provide a closed-form expression valid for all \( n > 1 \).

math.RA↗
arXiv2024

The $\mathfrak{sl}_2$-actions on the symmetric polynomials and on Young diagrams

In the article, two implementations of the representation of the complex Lie algebra $\mathfrak{sl}_2$ on the algebra of symmetric polynomials $Λ_n$ by differential operators are proposed. The realizations of irreducible subrepresentations, both finite-dimensional and infinite-dimensional, are described, and the decomposition of $Λ_n$ is found. The actions on the Schur polynomials is also determined. By using an isomorphism between $Λ_n$ and the vector space of Young diagrams $\mathbb{Q}\mathcal{Y}_n$ with no more than $n$ rows, these representations are transferred to $\mathbb{Q}\mathcal{Y}_n$.

math.CO↗
arXiv2020

3D geometric moment invariants from the point of view of the classical invariant theory

The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory. Using the remarkable fact that the groups $SO(3)$ and $SL(2)$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory. We give a precise statement of the 3D geometric invariant moments computation, introducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2)$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3)$ to an action of its Lie algebra $\mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition.

cs.CV↗
arXiv2019

Derivations and identities for Chebyshev polynomials of the first and second kinds

In this paper we follow the general approach, proposed earlier by the first author, which is derived from the invariant theory field and provides a way of obtaining of the polynomial identities for any arbitrary polynomial family. We introduce the notion of Chebyshev derivations of the first and second kinds, which is based on the polynomial algebra, and corresponding specific differential operators. We derive the elements of their kernels and prove that any element of the kernel of the derivations defines a polynomial identity satisfied by the Chebyshev polynomials of the first and second kinds. Combining elementary methods and combinatorial techniques, we obtain several new polynomial identities involving the Chebyshev polynomials of the both kinds and a special case of the Jacobi polynomials. Using the properties of the generalised hypergeometric function, we specify the Chebyshev polynomials of the first and second kinds via the generalised hypergeometric function and, as a consequence, derive the corresponding identities involving the generalised hypergeometric function and the Chebyshev polynomials of the first and second kinds.

math.CO↗
arXiv2019

2D moment invariants from the point of view of the classical invariant theory

Invariants allow to classify images up to the action of a group of transformations. In this paper we introduce notions of the algebras of simultaneous polynomial and rational 2D moment invariants and prove that they are isomorphic to the algebras of joint polynomial and rational $SO(2)$-invariants of binary forms. Also, to simplify the calculating of invariants we pass from an action of Lie group $SO(2)$ to an action of its Lie algebra $\mathfrak{so}_2$. This allow us to reduce the problem to standard problems of the classical invariant theory.

cs.CV↗
arXiv2018

The Double Star Sequences and the General Second Zagreb Index

For a simple graph we introduce notions of the double star sequence, the double star frequently sequence and prove that these sequences are inverses of each other. As a consequence, we express the general second Zagreb index in terms of the double star sequence. Also, we calculate the ordinary generating function and a linear recurrence relation for the sequence of the general second Zagreb indexes.

math.CO↗
arXiv2017

The star sequence and the general first Zagreb index

For a simple graph, we introduce a notion of the star sequence and prove that the star sequence and the frequently sequences of a graph are inverses of each other from a combinatorial point of view. As a consequence, we express the general first Zagreb index in terms of the star sequence. Also, we calculate the ordinary generating function and find a linear recurrence relation for the sequence of the general first Zagreb indexes.

math.CO↗