arXiv subjects

Mrinal Kumar

Publications and source records attributed to Mrinal Kumar.

At least 19 recordsLinked to original sources

arXiv2026

Deterministic factorization of constant depth circuits from whitebox PIT

We show that over the field of rational numbers, there is a deterministic algorithm for factorization of constant depth circuits that makes calls to a (whitebox or blackbox) PIT algorithm for such circuits. The time complexity of the algorithm (and hence the number of calls to the PIT oracle) is polynomially bounded in the size and degree of the input circuit. Our reduction to PIT is fairly simple and general and applies as it is to other natural classes of algebraic circuits like formulas and algebraic branching programs. Prior to this work, the known deterministic factorization algorithms for constant depth circuits crucially relied on \emph{blackbox} PIT [Bhattacharjee et al.]. Our main technical ingredient is an efficient, deterministic algorithm that, when given access to sufficiently high degree truncations of \emph{all} the power series roots of a constant depth circuit, uses oracle calls to any (whitebox or blackbox) PIT algorithm for these circuits, and partitions the roots into the sets of conjugates; thereby identifying the irreducible factors. For our proof, we rely on a root recombination criterion using logarithmic derivatives [Sasaki et al., Bostan et al., Lecerf] and show that this criterion can be efficiently and deterministically implemented for power series roots of constant depth circuits using access to any PIT oracle. On the way, we discuss an algorithm that may be of independent interest: it takes as input a matrix whose entries are multivariate constant depth circuits over a field $\mathbb{F}$, requires oracle access to any PIT algorithm for these circuits and outputs a basis for the $\mathbb{F}$-kernel of this matrix in polynomial time.

cs.CC↗

arXiv2026

Reed-Solomon Codes at Capacity: Algorithmic List-Decoding and Proximity Gaps

Understanding the limits of list-decodability of Reed-Solomon codes has been one of the most important open problems in algebraic coding theory. Recently, Brakensiek, Chen, Putterman, Zhang, and Zheng, in a remarkable breakthrough, showed that Reed-Solomon (RS) codes over fields of large characteristic are algorithmically list-decodable all the way up to capacity. Building on this result, Jeronimo subsequently extended these techniques to solve the proximity-gaps question for RS codes. In this article, we give a unified and transparent exposition of these results.

cs.CC↗

arXiv2026

A Quadratic Lower Bound on Determinantal Complexity

We prove an $Ω(n^2)$ lower bound on the determinantal complexity of the power sum polynomial $\sum_{i=1}^n x_i^n$ over the field of complex numbers. A similar result was claimed in a recent paper of Sheshadri (arXiv:2606.13628), via an AI-assisted and AI-written proof. Assuming its correctness, this was the first super-linear lower bound for this fundamental algebraic problem for any explicit polynomial. However, the authors of this note were unable to follow the details and verify the argument in arXiv:2606.13628, in spite of considerable effort on their part. The proof we provide here is short, (almost) self-contained and seemingly simpler.

cs.CC↗

arXiv2026

Refining GLSDC for Orbit Determination

This paper investigates the ambiguity of noisy short-arc angles-only orbit determination for a Molniya-orbit case study. Conventional Gauss Initial Orbit Determination followed by Gaussian Least-Squares Differential Correction is first shown to be unreliable under high-noise short-arc conditions, frequently failing to converge or converging to hyperbolic local solutions. To explore the candidate solution space, a Lambert-based initialization procedure is used over a grid of assumed ranges and angular-observation pairs, and the resulting states are refined using nonlinear least-squares optimization. The unconstrained solution set reveals multiple orbit families, including reentry elliptic, bounded elliptic, xGEO elliptic, and hyperbolic trajectories, all of which can reproduce the observed angular arc with comparable residuals. The solutions exhibit a structured range-velocity relationship, indicating that the angular measurements primarily constrain apparent line-of-sight motion rather than absolute range. Finally, physically motivated constraints are introduced to isolate Molniya-like bounded elliptic solutions. The results demonstrate that short-arc angles-only orbit determination is fundamentally non-unique without a priori information and that constraints should be interpreted as orbit-family selection tools rather than proof of uniqueness.

math.DS↗

arXiv2026

Multiclass Semantic Segmentation of Wildland Fire Images Using Context-Aware Centralized Copy-Paste Data Augmentation

Producing accurate annotations for deep learning based image segmentation is both costly and labor intensive. This challenge is especially evident in wildland fire applications, where accurately labeled datasets are scarce due to the difficulty of collecting and annotating dynamic fire scenes. To address this problem, our previous work introduced the Centralized Copy-Paste Data Augmentation (CCPDA) method for semantic segmentation of wildland fire imagery, which generates artificial training samples by randomly pasting fire clusters from source images onto target images. However, random placement can produce contextually unrealistic scenes, such as fire burning on asphalt. In this paper, we present a context-aware strategy designed specifically to improve data quality and realism in small multiclass wildland fire datasets, ensuring that augmented samples remain contextually meaningful. The proposed method restricts fire placement to semantically valid target regions and selects the location whose Ash-Vegetation composition most closely matches the source context. This approach preserves existing fire regions in the target image, prevents unrealistic placements, and maintains contextual accuracy by generating images that resemble real wildland fire scenes. We evaluate the Context-Aware CCPDA strategy through numerical analysis and comparisons with other augmentation methods by a weighted sum-based multi-objective optimization (MOO) approach. The results confirm that the context-aware data augmentation strategy leads to improved segmentation performance and contextual realism, outperforming other augmentation procedures.

cs.CV↗

arXiv2026

State Forecasting in an Estimation Framework with Surrogate Sensor Modeling

In recent years, computational power and data availability breakthroughs have revolutionized our ability to analyze complex physical systems through the inverse problem approach. Data-driven techniques like system identification and machine learning play an important role in this field, allowing us to gain insights into previously inaccessible phenomena. However, a major hurdle remains: How can meaningful information from partial measurements be extracted? In the aerospace domain, the challenge of state estimation is particularly pronounced due to the limited availability of observational data and the constraints imposed by sensor capabilities for tracking resident space objects (RSOs). To address these limitations, advanced compensation methodologies are required. Currently, range and bearing measurements obtained from radar and optical systems constitute the primary observational tools in the space situational awareness (SSA) community. In this work, we propose a novel framework that integrates a simplified reference dynamics model with a data-driven surrogate measurement model. This fusion process leverages the strengths of both models to estimate complex dynamical behaviors under conditions of partial observability. Extensive numerical experiments were conducted across multiple datasets to validate the proposed framework. The results demonstrate its efficacy in accurately reconstructing system dynamics from incomplete measurement data. Furthermore, to ensure the robustness of the framework, an initial consistency analysis of the surrogate modeling approach is presented. By addressing the current challenges and refining the integration of data-driven techniques with traditional physics-based modeling, this framework aims to advance state estimation methodologies in the aerospace sector.

eess.SY↗

arXiv2026

Centralized Copy-Paste: Enhanced Data Augmentation Strategy for Wildland Fire Semantic Segmentation

Collecting and annotating images for the purpose of training segmentation models is often cost prohibitive. In the domain of wildland fire science, this challenge is further compounded by the scarcity of reliable public datasets with labeled ground truth. This paper presents the Centralized Copy-Paste Data Augmentation (CCPDA) method, for the purpose of assisting with the training of deep-learning multiclass segmentation models, with special focus on improving segmentation outcomes for the fire-class. CCPDA has three main steps: (i) identify fire clusters in the source image, (ii) apply a centralization technique to focus on the core of the fire area, and (iii) paste the refined fire clusters onto a target image. This method increases dataset diversity while preserving the essential characteristics of the fire class. The effectiveness of this augmentation technique is demonstrated via numerical analysis and comparison against various other augmentation methods using a weighted sum-based multi-objective optimization approach. This approach helps elevate segmentation performance metrics specific to the fire class, which carries significantly more operational significance than other classes (fuel, ash, or background). Numerical performance assessment validates the efficacy of the presented CCPDA method in alleviating the difficulties associated with small, manually labeled training datasets. It also illustrates that CCPDA outperforms other augmentation strategies in the application scenario considered, particularly in improving fire-class segmentation performance.

cs.CV↗

arXiv2026

Deterministic list decoding of Reed-Solomon codes

We show that Reed-Solomon codes of dimension $k$ and block length $n$ over any finite field $\mathbb{F}$ can be deterministically list decoded from agreement $\sqrt{(k-1)n}$ in time $\text{poly}(n, \log |\mathbb{F}|)$. Prior to this work, the list decoding algorithms for Reed-Solomon codes, from the celebrated results of Sudan and Guruswami-Sudan, were either randomized with time complexity $\text{poly}(n, \log |\mathbb{F}|)$ or were deterministic with time complexity depending polynomially on the characteristic of the underlying field. In particular, over a prime field $\mathbb{F}$, no deterministic algorithms running in time $\text{poly}(n, \log |\mathbb{F}|)$ were known for this problem. Our main technical ingredient is a deterministic algorithm for solving the bivariate polynomial factorization instances that appear in the algorithm of Sudan and Guruswami-Sudan with only a $\text{poly}(\log |\mathbb{F}|)$ dependence on the field size in its time complexity for every finite field $\mathbb{F}$. While the question of obtaining efficient deterministic algorithms for polynomial factorization over finite fields is a fundamental open problem even for univariate polynomials of degree $2$, we show that additional information from the received word can be used to obtain such an algorithm for instances that appear in the course of list decoding Reed-Solomon codes.

cs.CC↗

arXiv2026

Advances in List Decoding of Polynomial Codes

Error-correcting codes are a method for representing data, so that one can recover the original information even if some parts of it were corrupted. The basic idea, which dates back to the revolutionary work of Shannon and Hamming about a century ago, is to encode the data into a redundant form, so that the original information can be decoded from the redundant encoding even in the presence of some noise or corruption. One prominent family of error-correcting codes are Reed-Solomon Codes which encode the data using evaluations of low-degree polynomials. Nearly six decades after they were introduced, Reed-Solomon Codes, as well as some related families of polynomial-based codes, continue to be widely studied, both from a theoretical perspective and from the point of view of applications. Besides their obvious use in communication, error-correcting codes such as Reed-Solomon Codes are also useful for various applications in theoretical computer science. These applications often require the ability to cope with many errors, much more than what is possible information-theoretically. List-decodable codes are a special class of error-correcting codes that enable correction from more errors than is traditionally possible by allowing a small list of candidate decodings. These codes have turned out to be extremely useful in various applications across theoretical computer science and coding theory. In recent years, there have been significant advances in list decoding of Reed-Solomon Codes and related families of polynomial-based codes. This includes efficient list decoding of such codes up to the information-theoretic capacity, with optimal list-size, and using fast nearly-linear time, and even sublinear-time, algorithms. In this book, we survey these developments.

cs.IT↗

arXiv2026

Generalized Parallel Scaling with Interdependent Generations

Parallel LLM inference scaling involves sampling a set of $N>1$ responses for a single input prompt. However, these $N$ parallel responses tend to be generated independently from each other, partitioning compute resources and leaving potentially useful information in one generation untapped by others. This is in contrast to response length scaling where past computation is used in all future steps. For higher quality responses and response sets, we propose Bridge to generate interdependent responses in parallel by rethinking batched LLM hidden states as holistic tensors rather than independent slices. With only a small amount (2.8%-5.1%) of new parameters, Bridge improves the relative mean accuracy gains from reinforcement learning with verifiable rewards by up to 39% and boosts consistency of correct responses. Trained once, Bridge scales to any generation width, all with greater performance than independent generations, unlocking a more general mode of parallel scaling that effectively leverages information between sequences, compatible with any post-generation aggregation technique.

cs.AI↗

arXiv2026

Modular composition & polynomial GCD in the border of small, shallow circuits

Modular composition is the problem of computing the coefficient vector of the polynomial $f(g(x)) \bmod h(x)$, given as input the coefficient vectors of univariate polynomials $f$, $g$, and $h$ over an underlying field $\mathbb{F}$. While this problem is known to be solvable in nearly-linear time over finite fields due to work of Kedlaya & Umans, no such near-linear-time algorithms are known over infinite fields, with the fastest known algorithm being from a recent work of Neiger, Salvy, Schost & Villard that takes $O(n^{1.43})$ field operations on inputs of degree $n$. In this work, we show that for any infinite field $\mathbb{F}$, modular composition is in the border of algebraic circuits with division gates of nearly-linear size and polylogarithmic depth. Moreover, this circuit family can itself be constructed in near-linear time. Our techniques also extend to other algebraic problems, most notably to the problem of computing greatest common divisors of univariate polynomials. We show that over any infinite field $\mathbb{F}$, the GCD of two univariate polynomials can be computed (piecewise) in the border sense by nearly-linear-size and polylogarithmic-depth algebraic circuits with division gates, where the circuits themselves can be constructed in near-linear time. While univariate polynomial GCD is known to be computable in near-linear time by the Knuth--Schönhage algorithm, or by constant-depth algebraic circuits from a recent result of Andrews & Wigderson, obtaining a parallel algorithm that simultaneously achieves polylogarithmic depth and near-linear work remains an open problem of great interest. Our result shows such an upper bound in the setting of border complexity.

cs.CC↗

arXiv2025

Algorithmizing the Multiplicity Schwartz-Zippel Lemma

The multiplicity Schwartz-Zippel lemma asserts that over a field, a low-degree polynomial cannot vanish with high multiplicity very often on a sufficiently large product set. Since its discovery in a work of Dvir, Kopparty, Saraf and Sudan [SIAM J. Comput., 2013], the lemma has found numerous applications in both math and computer science; in particular, in the definition and properties of multiplicity codes by Kopparty, Saraf and Yekhanin [J. ACM, 2014]. In this work, we show how to algorithmize the multiplicity Schwartz-Zippel lemma for arbitrary product sets over any field. In other words, we give an efficient algorithm for unique decoding of multivariate multiplicity codes from half their minimum distance on arbitrary product sets over all fields. Previously, such an algorithm was known either when the underlying product set had a nice algebraic structure: for instance, was a subfield (by Kopparty [ToC, 2015]) or when the underlying field had large (or zero) characteristic, the multiplicity parameter was sufficiently large and the multiplicity code had distance bounded away from $1$ (Bhandari, Harsha, Kumar and Sudan [STOC 2021]). In particular, even unique decoding of bivariate multiplicity codes with multiplicity two from half their minimum distance was not known over arbitrary product sets over any field. Our algorithm builds upon a result of Kim and Kopparty [ToC, 2017] who gave an algorithmic version of the Schwartz-Zippel lemma (without multiplicities) or equivalently, an efficient algorithm for unique decoding of Reed-Muller codes over arbitrary product sets. We introduce a refined notion of distance based on the multiplicity Schwartz-Zippel lemma and design a unique decoding algorithm for this distance measure. On the way, we give an alternate analysis of Forney's classical generalized minimum distance decoder that might be of independent interest.

cs.CC↗

arXiv2025

Fast list recovery of univariate multiplicity and folded Reed-Solomon codes

A recent work of Goyal, Harsha, Kumar and Shankar gave nearly linear time algorithms for the list decoding of Folded Reed-Solomon codes (FRS) and univariate multiplicity codes up to list decoding capacity in their natural setting of parameters. A curious aspect of this work was that unlike most list decoding algorithms for codes that also naturally extend to the problem of list recovery, the algorithm in the work of Goyal et al. seemed to be crucially tied to the problem of list decoding. In particular, it wasn't clear if their algorithm could be generalized to solve the problem of list recovery FRS and univariate multiplicity codes in near linear time. In this work, we address this question and design $\tilde{O}(n)$-time algorithms for list recovery of Folded Reed-Solomon codes and univariate Multiplicity codes up to capacity, where $n$ is the blocklength of the code. For our proof, we build upon the lattice based ideas crucially used by Goyal et al. with one additional technical ingredient - we show the construction of appropriately structured lattices over the univariate polynomial ring that \emph{capture} the list recovery problem for these codes.

cs.IT↗

arXiv2025

Temporally Consistent Koopman Autoencoders for Forecasting Dynamical Systems

Absence of sufficiently high-quality data often poses a key challenge in data-driven modeling of high-dimensional spatio-temporal dynamical systems. Koopman Autoencoders (KAEs) harness the expressivity of deep neural networks (DNNs), the dimension reduction capabilities of autoencoders, and the spectral properties of the Koopman operator to learn a reduced-order feature space with simpler, linear dynamics. However, the effectiveness of KAEs is hindered by limited and noisy training datasets, leading to poor generalizability. To address this, we introduce the temporally consistent Koopman autoencoder (tcKAE), designed to generate accurate long-term predictions even with limited and noisy training data. This is achieved through a consistency regularization term that enforces prediction coherence across different time steps, thus enhancing the robustness and generalizability of tcKAE over existing models. We provide analytical justification for this approach based on Koopman spectral theory and empirically demonstrate tcKAE's superior performance over state-of-the-art KAE models across a variety of test cases, including simple pendulum oscillations, kinetic plasma, and fluid flow data.

cs.LG↗

arXiv2025

Closure under factorization from a result of Furstenberg

We show that algebraic formulas and constant-depth circuits are closed under taking factors. In other words, we show that if a multivariate polynomial over a field of characteristic zero has a small constant-depth circuit or formula, then all its factors can be computed by small constant-depth circuits or formulas respectively. Our result turns out to be an elementary consequence of a fundamental and surprising result of Furstenberg from the 1960s, which gives a non-iterative description of the power series roots of a bivariate polynomial. Combined with standard structural ideas in algebraic complexity, we observe that this theorem yields the desired closure results. As applications, we get alternative (and perhaps simpler) proofs of various known results and strengthen the quantitative bounds in some of them. This includes a unified proof of known closure results for algebraic models (circuits, branching programs and VNP), an extension of the analysis of the Kabanets-Impagliazzo hitting set generator to formulas and constant-depth circuits, and a (significantly) simpler proof of correctness as well as stronger guarantees on the output in the subexponential time deterministic algorithm for factorization of constant-depth circuits from a recent work of Bhattacharjee, Kumar, Ramanathan, Saptharishi & Saraf.

cs.CC↗

arXiv2025

Constant-depth circuits for polynomial GCD over any characteristic

We show that the GCD of two univariate polynomials can be computed by (piece-wise) algebraic circuits of constant depth and polynomial size over any sufficiently large field, regardless of the characteristic. This extends a recent result of Andrews & Wigderson who showed such an upper bound over fields of zero or large characteristic. Our proofs are based on a recent work of Bhattacharjee, Kumar, Rai, Ramanathan, Saptharishi \& Saraf that shows closure of constant depth algebraic circuits under factorization. On our way to the proof, we show that any $n$-variate symmetric polynomial $P$ that has a small constant depth algebraic circuit can be written as the composition of a small constant depth algebraic circuit with elementary symmetric polynomials. This statement is a constant depth version of a result of Bläser & Jindal, who showed this for algebraic circuits of unbounded depth. As an application of our techniques, we also strengthen the closure results for factors of constant-depth circuits in the work of Bhattacharjee et al. over fields for small characteristic.

cs.CC↗

arXiv2025

Deterministic factorization of constant-depth algebraic circuits in subexponential time

While efficient randomized algorithms for factorization of polynomials given by algebraic circuits have been known for decades, obtaining an even slightly non-trivial deterministic algorithm for this problem has remained an open question of great interest. This is true even when the input algebraic circuit has additional structure, for instance, when it is a constant-depth circuit. Indeed, no efficient deterministic algorithms are known even for the seemingly easier problem of factoring sparse polynomials or even the problem of testing the irreducibility of sparse polynomials. In this work, we make progress on these questions: we design a deterministic algorithm that runs in subexponential time, and when given as input a constant-depth algebraic circuit $C$ over the field of rational numbers, it outputs algebraic circuits (of potentially unbounded depth) for all the irreducible factors of $C$, together with their multiplicities. In particular, we give the first subexponential time deterministic algorithm for factoring sparse polynomials. For our proofs, we rely on a finer understanding of the structure of power series roots of constant-depth circuits and the analysis of the Kabanets-Impagliazzo generator. In particular, we show that the Kabanets-Impagliazzo generator constructed using low-degree hard polynomials (explicitly constructed in the work of Limaye, Srinivasan & Tavenas) preserves not only the non-zeroness of small constant-depth circuits (as shown by Chou, Kumar & Solomon), but also their irreducibility and the irreducibility of their factors.

cs.CC↗

arXiv2025

Reinforcement Learning from User Feedback

As large language models (LLMs) are increasingly deployed in diverse user facing applications, aligning them with real user preferences becomes essential. Existing methods like Reinforcement Learning from Human Feedback (RLHF) rely on expert annotators trained on manually defined guidelines, whose judgments may not reflect the priorities of everyday users. We introduce Reinforcement Learning from User Feedback (RLUF), a framework for aligning LLMs directly to implicit signals from users in production. RLUF addresses key challenges of user feedback: user feedback is often binary (e.g., emoji reactions), sparse, and occasionally adversarial. We train a reward model, P[Love], to predict the likelihood that an LLM response will receive a Love Reaction, a lightweight form of positive user feedback, and integrate P[Love] into a multi-objective policy optimization framework alongside helpfulness and safety objectives. In large-scale experiments, we show that P[Love] is predictive of increased positive feedback and serves as a reliable offline evaluator of future user behavior. Policy optimization using P[Love] significantly raises observed positive-feedback rates, including a 28% increase in Love Reactions during live A/B tests. However, optimizing for positive reactions introduces reward hacking challenges, requiring careful balancing of objectives. By directly leveraging implicit signals from users, RLUF offers a path to aligning LLMs with real-world user preferences at scale.

cs.AI↗