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Sanjay Kumar Singh

Publications and source records attributed to Sanjay Kumar Singh.

At least 19 recordsLinked to original sources
arXiv2026

Construction of codes over a commutative non-unital ring from simplicial complexes and their applications

In this article, we investigate the construction of linear codes over a finite ring $\mathcal{S}$, where $\mathcal{S}$ is taken to be an extension of a commutative non-unital ring $I$ of order $p^2$. Our approach is based on the defining set method. The defining sets considered in this work are derived from general simplicial complexes that may contain multiple maximal elements. We determine the parameters of these codes over $\mathcal{S}$ and study their Gray images. We also study the corresponding subfield-like codes. We show that these Gray image codes and subfield-like codes produce several families of divisible codes. Furthermore, we establish sufficient conditions under which these codes are minimal, optimal, and self-orthogonal. As applications of our results, we obtain several families of projective few-weight codes, and locally recoverable codes with small locality. We also study the minimal access structures of secret-sharing schemes associated with the duals of these minimal codes. Moreover, we construct several families of strongly regular graphs from projective two-weight codes and determine their parameters explicitly.

cs.IT↗
arXiv2026

Gauss sum with principal multiplicative character

Let $R$ be a finite ring with unity, $ψ: R \to \mathbb{C}^\times$ be an additive character of $R$, and \( χ_0 \) be the principal multiplicative character ($i.e.$, $χ_0(x) = 1 \quad \text{for all } x \in R^\times$), then the Gauss sum is \[ G(χ_0, ψ) = \sum_{x \in R^\times} ψ(x). \] In this paper, we give an explicit formula for a more general form of the Gauss sum $G(χ_0, ψ)$. Interestingly, the formula extends the known formula of classical Ramanujan's sum to the context of finite rings. As an application, we derive the eigenvalues for a more general form of the unitary Cayley graph $\text{Cay}(R, R^{\times})$ using the formula.

math.CO↗
arXiv2026

Nonexistence results of generalized bent functions from $\mathbb{Z}_3^n$ to $ \mathbb{Z}_m$

In this paper, we investigate generalized bent functions (GBFs) from $\mathbb{Z}_3^n$ to $\mathbb{Z}_m$. We show that GBFs exist whenever $3$ divides $m$, while several nonexistence results are obtained when $3\nmid m$. In particular, we prove that no GBFs exist for $n=1,2$ when $m$ is odd and not divisible by $3$. For the case $n=3$, we establish the nonexistence of GBFs $f:\mathbb{Z}_3^3 \rightarrow \mathbb{Z}_{5\cdot11^r}$ for all nonnegative integers $r$. Finally, we show that no GBF exists from $\mathbb{Z}_3$ to $\mathbb{Z}_{2m'}$ and $\mathbb{Z}_3^2$ to $\mathbb{Z}_{2m'}$, where $m'$ is odd and not divisible by $3$.

math.CO↗
arXiv2026

Detection Without Correction: A Robust Asymmetry in Activation-Based Hallucination Probing

Activation-based linear probing is widely proposed as a method for both detecting and correcting hallucinations in autoregressive language models. We present an empirical study across seven models spanning 117M to 7B parameters and three architecture families (GPT-2, Pythia, Qwen-2.5) that documents a robust asymmetry: linear probes can detect hallucination signals with above-chance accuracy in larger models, but activation steering along the probe-derived direction fails to correct hallucinations in 7 of 7 models tested. We further find that output-confidence baselines outperform activation probes on raw detection AUC at every model above 410M parameters, with the gap reaching 0.157 AUC for Pythia-6.9B. The probe's distinguishing value is therefore not detection accuracy but temporal positioning: probe signals are accessible at position zero (before any output tokens are produced), enabling pre-generation flagging that output-based methods structurally cannot provide. The temporal signal is statistically significant in two of seven models (Pythia-1.4B, p = 0.012; Qwen2.5-7B, p = 0.038) and absent in models below 400M parameters and in the base-only Pythia-6.9B. We position these findings as a clean negative result for the dominant probing-as-detection-and-control research direction and as initial evidence that probe-based methods occupy a complementary deployment niche, namely pre-generation flagging, rather than competing with output-based detectors on raw accuracy.

cs.CL↗
arXiv2026

A Multi-Level Causal Intervention Framework for Mechanistic Interpretability in Variational Autoencoders

Understanding how generative models represent and transform data is a foundational problem in deep learning interpretability. While mechanistic interpretability of discriminative architectures has yielded substantial insights, relatively little work has addressed variational autoencoders (VAEs). This paper presents the first general-purpose multilevel causal intervention framework for mechanistic interpretability of VAEs. The framework comprises four manipulation types: input manipulation, latent-space perturbation, activation patching, and causal mediation analysis. We also define three new quantitative metrics capturing properties not measured by existing disentanglement metrics alone: Causal Effect Strength (CES), intervention specificity, and circuit modularity. We conduct the largest empirical study to date of VAE causal mechanisms across six architectures (standard VAE, beta-VAE, FactorVAE, beta-TC-VAE, DIP-VAE-II, and VQ-VAE) and five benchmarks (dSprites, 3DShapes, MPI3D, CelebA, and SmallNORB), with three seeds per configuration, totaling 90 independent training runs. Our results reveal several findings: (i) a consistent within-dataset negative correlation between CES and DCI disentanglement (the CES-DCI trade-off); (ii) that the KL reweighting mechanism of beta-VAE induces a capacity bottleneck when generative factors approach latent dimensionality, degrading disentanglement on complex datasets; (iii) that no single VAE architecture dominates across all five datasets, with optimal choice depending on dataset structure; and (iv) that CES-based metrics applied to discrete latent spaces (VQ-VAE) yield near-zero values, revealing a critical limitation of continuous-intervention methods for discrete representations. These results provide both a theoretical foundation and comprehensive empirical evaluation for mechanistic interpretability of generative models.

cs.LG↗
arXiv2026

Posterior-Calibrated Causal Circuits in Variational Autoencoders: Why Image-Domain Interpretability Fails on Tabular Data

Although mechanism-based interpretability has generated an abundance of insight for discriminative network analysis, generative models are less understood -- particularly outside of image-related applications. We investigate how much of the causal circuitry found within image-related variational autoencoders (VAEs) will generalize to tabular data, as VAEs are increasingly used for imputation, anomaly detection, and synthetic data generation. In addition to extending a four-level causal intervention framework to four tabular and one image benchmark across five different VAE architectures (with 75 individual training runs per architecture and three random seed values for each run), this paper introduces three new techniques: posterior-calibration of Causal Effect Strength (CES), path-specific activation patching, and Feature-Group Disentanglement (FGD). The results from our experiments demonstrate that: (i) Tabular VAEs have circuits with modularity that is approximately 50% lower than their image counterparts. (ii) $β$-VAE experiences nearly complete collapse in CES scores when applied to heterogeneous tabular features (0.043 CES score for tabular data compared to 0.133 CES score for images), which can be directly attributed to reconstruction quality degradation (r = -0.886 correlation coefficient between CES and MSE). (iii) CES successfully captures nine of eleven statistically significant architecture differences using Holm--Šidák corrections. (iv) Interventions with high specificity predict the highest downstream AUC values (r = 0.460, p < .001). This study challenges the common assumption that architectural guidance from image-related studies can be transferred to tabular datasets.

cs.LG↗
arXiv2026

Fundamental Limits of Neural Network Sparsification: Evidence from Catastrophic Interpretability Collapse

Extreme neural network sparsification (90% activation reduction) presents a critical challenge for mechanistic interpretability: understanding whether interpretable features survive aggressive compression. This work investigates feature survival under severe capacity constraints in hybrid Variational Autoencoder--Sparse Autoencoder (VAE-SAE) architectures. We introduce an adaptive sparsity scheduling framework that progressively reduces active neurons from 500 to 50 over 50 training epochs, and provide empirical evidence for fundamental limits of the sparsification-interpretability relationship. Testing across two benchmark datasets -- dSprites and Shapes3D -- with both Top-k and L1 sparsification methods, our key finding reveals a pervasive paradox: while global representation quality (measured by Mutual Information Gap) remains stable, local feature interpretability collapses systematically. Under Top-k sparsification, dead neuron rates reach $34.4\pm0.9\%$ on dSprites and $62.7\pm1.3\%$ on Shapes3D at k=50. L1 regularization -- a fundamentally different "soft constraint" paradigm -- produces equal or worse collapse: $41.7\pm4.4\%$ on dSprites and $90.6\pm0.5\%$ on Shapes3D. Extended training for 100 additional epochs fails to recover dead neurons, and the collapse pattern is robust across all tested threshold definitions. Critically, the collapse scales with dataset complexity: Shapes3D (RGB, 6 factors) shows $1.8\times$ more dead neurons than dSprites (grayscale, 5 factors) under Top-k and $2.2\times$ under L1. These findings establish that interpretability collapse under sparsification is intrinsic to the compression process rather than an artifact of any particular algorithm, training duration, or threshold choice.

cs.LG↗
arXiv2025

A generalization of Ramanujan's sum over finite groups

Let $G$ be a finite group, and let $x \in G$. Define $[x^G] := \{ y \in G : \langle x^G \rangle = \langle y^G \rangle \}$, where $\langle x^G \rangle$ denotes the normal subgroup of $G$ generated by the conjugacy class of $x$. In this paper, we determine an explicit formula for the eigenvalues of the normal Cayley graph $\text{Cay}(G, [x^G])$. These eigenvalues can be viewed as a generalization of classical Ramanujan's sum in the setting of finite groups. Surprisingly, the formula we derive for the eigenvalues of $\text{Cay}(G, [x^G])$ extends the known formula of classical Ramanujan's sum to the context of finite groups. This generalization not only enrich the theory of Ramanujan's sum but also provide new tools in spectral graph theory, representation theory, and algebraic number theory.

math.NT↗
arXiv2025

Spectral properties of Cayley graphs over finite commutative rings

Let $R$ be a finite commutative ring with unity and $x$ be a non-zero element of $R$. In this paper, we calculate the spectrum and energy of the Cayley graph ${\rm Cay}(R,xR^{*})$, and also compute the energy of their compliment graph. Further, we give necessary and sufficient condition for Cayley graph ${\rm Cay}(R,xR^{*})$ to be Ramanujan.

math.CO↗
arXiv2023

On Certain Gluing of semigroup rings and indispensable resolution of semigroup rings

In this paper, our aim is twofold: First, by using the technique of gluing semigroups, we give infinitely many families of a projective closure with the Cohen-Macaulay (Gorenstein) property. Also, we give an effective technique for constructing large families of one dimensional Gorenstein local rings associated to monomial curves, which supports Rossi question, saying that every Gorenstein local ring has a non-decreasing Hilbert function. In the second part, we study strong indispensable minimal free resolutions of semigroup rings, focusing on the operation of the join of affine semigroups, which provide class of examples supporting Charalambous and Thoma question on the class of lattice ideal which has a strong indispensable free resolution.

math.AC↗
arXiv2021

Second Order parallel tensor on generalized f.pk-space form and hypersurfaces of generalized f.pk-space form

The purpose of the present paper to study a second order symmetric parallel tensor in generalized f.pk-space form. Second order symmetric parallel tensor in f.pk-space form is combination of the associated metric tensor and $1$-forms of structure vector fields. We prove that there does not exist second order skew-symmetric parallel tensor in f.pk-space form. We also deduce that there is no parallel hypersurface in a generalized f.pk-space form but there is semi-parallel hypersurfaces in a generalized f.pk-space form.

math.DG↗
arXiv2021

Comprehensive quasi-Einstein spacetime with application to general relativity

The aim of this paper is to extend the notion of all known quasi-Einstein manifolds like generalized quasi-Einstein, mixed generalized quasi-Einstein manifold, pseudo generalized quasi-Einstein manifold and many more and name it comprehensive quasi Einstein manifold C(QE)$_{n}$. We investigate some geometric and physical properties of the comprehensive quasi Einstein manifolds C(QE)$_{n}$ under certain conditions. We study the conformal and conharmonic mappings between C(QE)$_{n}$ manifolds. Then we examine the C(QE)$_{n}$ with harmonic Weyl tensor. We investigate geometric and physical properties of the comprehensive quasi Einstein manifolds C(QE)$_{n}$ under certain conditions. We define the manifold of comprehensive quasi-constant curvature and proved that conformally flat C(QE)$_{n}$ is manifold of comprehensive quasi-constant curvature and vice versa. We study the general two viscous fluid spacetime C(QE)$_{4}$ and find out some important consequences about C(QE)$_{4}$. We study C(QE)$_{n}$ with vanishing space matter tensor. Finally, we prove the existence of such manifolds by constructing non-trivial example.

math.DG↗
arXiv2021

Generalization of trace codes to places of higher degree

In this note, we give a construction of codes on algebraic function field $F/ \mathbb{F}_{q}$ using places of $F$ (not necessarily of degree one) and trace functions from various extensions of $\mathbb{F}_{q}$. This is a generalization of trace code of geometric Goppa codes to higher degree places. We compute a bound on the dimension of this code. Furthermore, we give a condition under which we get exact dimension of the code. We also determine a bound on the minimum distance of this code in terms of $B_{r}(F)$ ( the number of places of degree $r$ in $F$), $1 \leq r < \infty$. Few quasi-cyclic codes over $\mathbb{F}_{p}$ are also obtained as examples of these codes.

cs.IT↗
arXiv2020

A Note on Parabolic Bundles on Nodal Curves

Mehta and Seshadri have proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this note, we discuss the Mehta-Seshadri correspondence over an irreducible projective curve with at most nodes as singularities.

math.AG↗
arXiv2020

Projective Poincaré and Picard bundles for moduli spaces of vector bundles over nodal curves

Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L(n,d)$ of semistable vector bundles of rank $n$ with determinant $L$ of degree $d$ is isomorphic to $\mathbb{Z}$.

math.AG↗
arXiv2020

Generalized Hamming weights of toric codes over hypersimplices and square-free affine evaluation codes

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements, where $q$ is a power of prime $p$. A polynomial over $\mathbb{F}_{q}$ is square-free if all its monomials are square-free. In this note, we determine an upper bound on the number of zeroes in the affine torus $T=(\mathbb{F}_{q}^{*})^{s}$ of any set of $r$ linearly independent square-free polynomials over $\mathbb{F}_{q}$ in $s$ variables, under certain conditions on $r$, $s$ and degree of these polynomials. Applying the results, we partly obtain the generalized Hamming weights of toric codes over hypersimplices and square-free evaluation codes, as defined in \cite{hyper}. Finally, we obtain the dual of these toric codes with respect to the Euclidean scalar product.

math.AC↗
arXiv2020

On geometric Goppa codes from Elementary Abelian $p$-Extensions of $\mathbb{F}_{p^{s}}(x)$

Let $p$ be a prime number and $s> 0$ an integer. In this short note, we investigate one-point geometric Goppa codes associated with an elementary abelian $p$-extension of $\mathbb{F}_{p^{s}}(x)$. We determine their dimension and the exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list the exact values of the second generalized Hamming weight of these codes in a few cases. Simple criteria for the self-duality and the quasi-self-duality of these codes are also provided. Furthermore, we construct examples of quantum codes, convolutional codes, and locally recoverable codes on the function field.

cs.IT↗