arXiv subjects

Vinay Shukla

Publications and source records attributed to Vinay Shukla.

11 recordsLinked to original sources

arXiv2026

Strong asymptotics for Jacobi-Pi\~neiro orthogonal polynomials

We investigate the asymptotic behavior of Jacobi-Pi\~neiro polynomials of degree $2n$ orthogonal on $[0,1]$ with respect to weights $w_j(x) = x^{\alpha_j}(1-x)^{\beta}$, $j=1,2$ where $\alpha_1,\alpha_2, \beta>-1$, and $\alpha_1-\alpha_2 \notin \mathbb{Z}$. These polynomials are characterized by a Riemann-Hilbert problem for a $3 \times 3$ matrix-valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics in the complex plane. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

math.CA↗

arXiv2026

Evaluation of terminating and non-terminating sums containing the digamma function

We derive transformation and summation formulas for terminating and nonterminating series involving the digamma function. Our principal results are obtained by a limiting process starting with duality relations for the generalized hypergeometric functions and their consequences. Selected formulas are further extended by parameter differentiation of Euler's transformation and by using contiguous relations. Most of our identities express products of hypergeometric and digamma series in terms of hypergeometric functions, and some evaluations of terminating digamma sums involve Bernoulli polynomials. In several cases the digamma contributions cancel, producing identities involving only products of hypergeometric functions.

math.CA↗

arXiv2026

On Askey's extension of Clausen's identity and its polynomial perturbation

The celebrated Clausen's identity expresses the square of the Gauss hypergeometric series ${}_2F_{1}(a,b;a+b+1/2;x)$ as a single hypergeometric ${}_3F_2$ series. Goursat showed in 1883 that replacing $1/2$ by $m+1/2$ leads to a hypergeometric series for the square whenever $m$ is a positive integer. Askey found this series explicitly for $m=1$. The first goal of this paper is to extend this result by treating the case of any natural $m$. The ${}_3F_{2}$ series on the right-hand side is thereby replaced by its perturbation by an explicit characteristic polynomial of degree $2m$, i.e., its coefficients are multiplied by values of this polynomial at nonnegative integers. The second goal of this paper is to make one further step and replace the square of the Gauss function by its product with its perturbation by an arbitrary polynomial of degree $s\le{2m+1}$. We show that such product remains hypergeometric and find its explicit form in terms of a polynomial perturbation of the ${}_3F_2$ series. We present an explicit formula for the characteristic polynomial whose degree is shown to be $2m+s$.

math.CA↗

arXiv2025

Dual-Head Reasoning Distillation: Improving Classifier Accuracy with Train-Time-Only Reasoning

Chain-of-Thought (CoT) prompting often improves classification accuracy, but it introduces a significant throughput penalty with rationale generation (Wei et al., 2022; Cheng and Van Durme, 2024). To resolve this trade-off, we introduce Dual-Head Reasoning Distillation (DHRD), a simple training method for decoder-only language models (LMs) that adds (i) a pooled classification head used during training and inference and (ii) a reasoning head supervised by teacher rationales used only in training. We train with a loss function that is a weighted sum of label cross-entropy and token-level LM loss over input-plus-rationale sequences. On seven SuperGLUE tasks, DHRD yields relative gains of 0.65-5.47% over pooled baselines, with notably larger gains on entailment/causal tasks. Since we disable the reasoning head at test time, inference throughput matches pooled classifiers and exceeds CoT decoding on the same backbones by 96-142 times in QPS.

cs.CL↗

arXiv2025

WaterFlow: Learning Fast & Robust Watermarks using Stable Diffusion

The ability to embed watermarks in images is a fundamental problem of interest for computer vision, and is exacerbated by the rapid rise of generated imagery in recent times. Current state-of-the-art techniques suffer from computational and statistical challenges such as the slow execution speed for practical deployments. In addition, other works trade off fast watermarking speeds but suffer greatly in their robustness or perceptual quality. In this work, we propose WaterFlow (WF), a fast and extremely robust approach for high fidelity visual watermarking based on a learned latent-dependent watermark. Our approach utilizes a pretrained latent diffusion model to encode an arbitrary image into a latent space and produces a learned watermark that is then planted into the Fourier Domain of the latent. The transformation is specified via invertible flow layers that enhance the expressivity of the latent space of the pre-trained model to better preserve image quality while permitting robust and tractable detection. Most notably, WaterFlow demonstrates state-of-the-art performance on general robustness and is the first method capable of effectively defending against difficult combination attacks. We validate our findings on three widely used real and generated datasets: MS-COCO, DiffusionDB, and WikiArt.

eess.IV↗

arXiv2024

Stability of the Toda equations related to a perturbed $R_i$ type recurrence relation

In this manuscript, a modified $R_I$ type recurrence relation is considered whose recurrence coefficients are perturbed by addition or multiplication of a constant. The perturbed system of recurrence coefficients is represented by Toda lattice equations, which are derived. These equations are then represented in a matrix form. With the help of this matrix representation, a known Lax pair is recovered. Inferences about the stability of resulting perturbed system of Toda equations are drawn based on numerical experiments.

math.CA↗

arXiv2024

Spectral transformation associated with a perturbed $R_I$ type recurrence relation

In this work, orthogonal polynomials satisfying $R_I$ type recurrence relation %$\mathcal{P}_{n+1}(z) = (z-c_n)\mathcal{P}_n(z)-λ_n (z-a_n)\mathcal{P}_{n-1}(z),$ with $\mathcal{P}_{-1}(z) = 0$ and $\mathcal{P}_0(z) = 1$ are analyzed when the recurrence coefficients are modified. The structural relationship between the perturbed and the unperturbed polynomials along with the spectral properties and spectral transformation of continued fraction are investigated. It is demonstrated that the transfer matrix method is computationally more efficient than the classical method for obtaining perturbed $R_I$ polynomials. Further, an interesting consequence of co-dilation on the Carathéodary function is presented. Finally, the study of co-recursion and co-dilation in connection to the unit circle is carried out with the help of an illustration. The interlacing and monotonicity of zeros between L-Jacobi polynomials and their perturbed forms are demonstrated.

math.CA↗

arXiv2024

Generalized co-polynomials of $R_{II}$ type and associated quadrature rules

When the co-recursion and co-dilation in the recurrence relation of certain sequences of orthogonal polynomials are not at the same level, the behaviour of the modified orthogonal polynomials is expected to have different properties compared to the situation of the same level of perturbation. This manuscript attempts to derive structural relations between the perturbed and original $R_{II}$ type orthogonal polynomials. The classical result is improved using a transfer matrix approach. It turns out that the $R_{II}$ fraction with perturbation is the rational spectral transformation of the unperturbed one. The derived notions are used to deduce some consequences for the polynomials orthogonal on the real line. A natural question that arises while dealing with perturbations at different levels, i.e., which perturbation, co-recursion or co-dilation, needs to be performed first, is answered.

math.CA↗

arXiv2023

A Unified Approach to Count-Based Weakly-Supervised Learning

High-quality labels are often very scarce, whereas unlabeled data with inferred weak labels occurs more naturally. In many cases, these weak labels dictate the frequency of each respective class over a set of instances. In this paper, we develop a unified approach to learning from such weakly-labeled data, which we call count-based weakly-supervised learning. At the heart of our approach is the ability to compute the probability of exactly k out of n outputs being set to true. This computation is differentiable, exact, and efficient. Building upon the previous computation, we derive a count loss penalizing the model for deviations in its distribution from an arithmetic constraint defined over label counts. We evaluate our approach on three common weakly-supervised learning paradigms and observe that our proposed approach achieves state-of-the-art or highly competitive results across all three of the paradigms.

cs.LG↗

arXiv2022

Spectral properties related to generalized complementary Romanovski-Routh polynomials

Complementary Romanovski-Routh polynomials play an important role in extracting specific properties of orthogonal polynomials. In this work, a generalized form of the Complementary Romanovski-Routh polynomials (GCRR) that has the Gaussian hypergeometric representation and satisfies a particular type of recurrence called $R_{II}$ type three term recurrence relation involving two arbitrary parameters is considered. Self perturbation of GCRR polynomials leading to extracting two different types of $R_{II}$ type orthogonal polynomials are identified. Spectral properties of these resultant polynomials in terms of tri-diagonal linear pencil were analyzed. The LU decomposition of these pencil matrices provided interesting properties involving biorthogonality. Interlacing properties between the zeros of the polynomials in the discussion are established.

math.CA↗

arXiv2022

Chain sequences and Zeros of a perturbed $R_{II}$ type recurrence relation

In this manuscript, new algebraic and analytic aspects of the orthogonal polynomials satisfying $R_{II}$ type recurrence relation given by \begin{align*} \mathcal{P}_{n+1}(x) = (x-c_n)\mathcal{P}_n(x)-λ_n (x-a_n)(x-b_n)\mathcal{P}_{n-1}(x), \quad n \geq 0, \end{align*} where $λ_n$ is a positive chain sequence and $a_n$, $b_n$, $c_n$ are sequences of real or complex numbers with $\mathcal{P}_{-1}(x) = 0$ and $\mathcal{P}_0(x) = 1$ are investigated when the recurrence coefficients are perturbed. Specifically, representation of new perturbed polynomials (co-polynomials of $R_{II}$ type) in terms of original ones with the interlacing and monotonicity properties of zeros are given. For finite perturbations, a transfer matrix approach is used to obtain new structural relations. Effect of co-dilation in the corresponding chain sequences and their consequences onto the unit circle are analysed. A particular perturbation in the corresponding chain sequence called complementary chain sequences and its effect on the corresponding Verblunsky coefficients is also studied.

math.CA↗