arXiv subjects

Justin Zhang

Publications and source records attributed to Justin Zhang.

13 recordsLinked to original sources

arXiv2025

Non-negligible summands in tensor powers of some modular representations of finite $p$-groups

Let $p>0$ be a prime, $G$ be a finite $p$-group and $\Bbbk$ be an algebraically closed field of characteristic $p$. Dave Benson has conjectured that if $p=2$ and $V$ is an odd-dimensional indecomposable representation of $G$ then all summands of the tensor product $V \otimes V^*$ except for $\Bbbk$ have even dimension. It is known that the analogous result for general $p$ is false. In this paper, we investigate the class of graded representations $V$ which have dimension coprime to $p$ and for which $V \otimes V^*$ has a non-trivial summand of dimension coprime to $p$, for a graded group scheme closely related to $\mathbb{Z}/p^r \mathbb{Z} \times \mathbb{Z}/p^s \mathbb{Z}$, where $r$ and $s$ are nonnegative integers and $p>2$. We produce an infinite family of such representations in characteristic 3 and show in particular that the tensor subcategory generated by any of these representations in the semisimplification contains the modulo $3$ reduction of the category of representations of the symmetric group $S_3$. Our results are compatible with a general version of Benson's conjecture due to Etingof.

math.RT↗

arXiv2025

Amortized Locally Decodable Codes for Insertions and Deletions

Locally Decodable Codes (LDCs) are error correcting codes which permit the recovery of any single message symbol with a low number of queries to the codeword (the locality). Traditional LDC tradeoffs between the rate, locality, and error tolerance are undesirable even in relaxed settings where the encoder/decoder share randomness or where the channel is resource-bounded. Recent work by Blocki and Zhang initiated the study of Hamming amortized Locally Decodable Codes (aLDCs), which allow the local decoder to amortize their number of queries over the recovery of a small subset of message symbols. Surprisingly, Blocki and Zhang construct asymptotically ideal (constant rate, constant amortized locality, and constant error tolerance) Hamming aLDCs in private-key and resource-bounded settings. While this result overcame previous barriers and impossibility results for Hamming LDCs, it is not clear whether the techniques extend to Insdel LDCs. Constructing Insdel LDCs which are resilient to insertion and/or deletion errors is known to be even more challenging. Our first contribution is to provide a Hamming-to-Insdel compiler which transforms any amortized Hamming LDC that satisfies a particular property to amortized Insdel LDC while asymptotically preserving the rate, error tolerance and amortized locality. Prior Hamming-to-Insdel compilers worked for arbitrary Hamming LDCs, but incurred an undesirable polylogarithmic blow-up in the locality. Our second contribution is a construction of an ideal amortized Hamming LDC which satisfies our special property in the relaxed settings where the sender/receiver share randomness or where the channel is resource bounded. Taken together, we obtain ideal Insdel aLDCs in private-key and resource-bounded settings with constant amortized locality, constant rate and constant error tolerance.

cs.IT↗

arXiv2025

PuzzleWorld: A Benchmark for Multimodal, Open-Ended Reasoning in Puzzlehunts

Puzzlehunts are a genre of complex, multi-step puzzles lacking well-defined problem definitions. In contrast to conventional reasoning benchmarks consisting of tasks with clear instructions and constrained environments, puzzlehunts requires discovering the underlying problem structure from multimodal evidence and iterative reasoning, mirroring real-world domains such as scientific discovery, exploratory data analysis, or investigative problem-solving. Despite progress in foundation models, their performance on open-ended settings remains largely untested. We introduce PuzzleWorld, a comprehensive benchmark of 667 puzzlehunt-style problems designed to assess step-by-step, open-ended, and creative multimodal reasoning. Each puzzle is annotated with the final solution, detailed reasoning traces, and cognitive skill labels, enabling holistic benchmarking and fine-grained diagnostic analysis. Most state-of-the-art models achieve only 1-4% final answer accuracy. On PuzzleWorld, the best model solves only 18% of puzzles and reaches 40% stepwise accuracy, matching human puzzle novices but falling significantly behind puzzle enthusiasts. To demonstrate the value of our reasoning annotations, we show that fine-tuning a small model on reasoning traces boosts stepwise accuracy from 4% to 11%, which translates to improvements in downstream visual reasoning tasks. Our detailed error analysis reveals that current models exhibit myopic reasoning, are bottlenecked by the limitations of language-based inference, and lack sketching capabilities crucial for visual and spatial reasoning. We release PuzzleWorld at https://github.com/MIT-MI/PuzzleWorld to support future work on building more general, open-ended, and creative reasoning systems.

cs.CL↗

arXiv2025

Amortized Locally Decodable Codes

Locally Decodable Codes (LDCs) are error correcting codes that admit efficient decoding of individual message symbols without decoding the entire message. Unfortunately, known LDC constructions offer a sub-optimal trade-off between rate, error tolerance and locality, the number of queries that the decoder must make to the received codeword $\tilde {y}$ to recovered a particular symbol from the original message $x$, even in relaxed settings where the encoder/decoder share randomness or where the channel is resource bounded. We initiate the study of Amortized Locally Decodable Codes where the local decoder wants to recover multiple symbols of the original message $x$ and the total number of queries to the received codeword $y$ can be amortized by the total number of message symbols recovered. We demonstrate that amortization allows us to overcome prior barriers and impossibility results. We first demonstrate that the Hadamard code achieves amortized locality below $2$ -- a result that is known to be impossible without amortization. Second, we study amortized locally decodable codes in cryptographic settings where the sender and receiver share a secret key or where the channel is resource-bounded and where the decoder wants to recover a consecutive subset of message symbols $[L,R]$. In these settings we show that it is possible to achieve a trifecta: constant rate, error tolerance and constant amortized locality.

cs.IT↗

arXiv2024

PHICOIN (PHI): The Proof of Work High-Performance Infrastructure

PHICOIN (PHI) is a high-performance cryptocurrency based on the Proof-of-Work (PoW) mechanism. It aims to provide ordinary users with decentralized participation opportunities through an improved and innovative mining algorithm and fair design principles. PHI addresses the challenges of centralization in cryptocurrency mining by enhancing resistance to ASIC and FPGA devices and promoting fair participation. This paper outlines the technical specifications, mission, and roadmap for PHI, highlighting its potential to become a foundational infrastructure for PoW cryptocurrencies.

cs.CR↗

arXiv2023

The Ideal of Vanishing Polynomials and the Ring of Polynomial Functions

Vanishing polynomials are polynomials over a ring which output $0$ for all elements in the ring. In this paper, we study the ideal of vanishing polynomials over specific types of rings, along with the closely related ring of polynomial functions. In particular, we provide several results on generating vanishing polynomials. We first analyze the ideal of vanishing polynomial over $\mathbb{Z}_n$, the ring of the integers modulo $n$. We then establish an isomorphism between the vanishing polynomials of a ring and the vanishing polynomials of the constituent rings in its decomposition. Lastly, we generalize our results to study the ideal of vanishing polynomials over arbitrary commutative rings.

math.AC↗

arXiv2023

Experimental Results on Potential Markov Partitions for Wang Shifts

In this article we discuss potential Markov partitions for three different Wang tile protosets. The first partition is for the order-24 aperiodic Wang tile protoset that was recently shown in the Ph.D. thesis of H. Jang to encode all tilings by the Penrose rhombs. The second is a partition for an order-16 aperiodic Wang protoset that encodes all tilings by the Ammann A2 aperiodic protoset. The third partition is for an order-11 Wang tile protoset identified by Jeandel and Rao as a candidate order-11 aperiodic Wang tile protoset. The emphasis is on some experimental methodology to generate potential Markov partitions that encode tilings. We also apply some of the theory developed by Labb\'{e} in analyzing such an experimentally discovered partition.

math.DS↗

arXiv2023

Results on Vanishing Polynomials and Polynomial Root Counting

We study the set of algebraic objects known as vanishing polynomials (the set of polynomials that annihilate all elements of a ring) over general commutative rings with identity. These objects are of special interest due to their close connections to both ring theory and the technical applications of polynomials, along with numerous applications to other mathematical and engineering fields. We first determine the minimum degree of monic vanishing polynomials over a specific infinite family of rings of a specific form and consider a generalization of the notion of a monic vanishing polynomial over a subring. We then present a partial classification of the ideal of vanishing polynomials over general commutative rings with identity of prime and prime square orders. Finally, we prove some results on rings that have a finite number of roots and propose a technique that can be utilized to restrict the number of roots polynomials can have over certain finite commutative rings.

math.AC↗

arXiv2022

Prediction of Oral Food Challenge Outcomes via Ensemble Learning

Oral Food Challenges (OFCs) are essential to accurately diagnosing food allergy due to the limitations of existing clinical testing. However, some patients are hesitant to undergo OFCs, while those willing suffer from limited access to allergists in rural/community healthcare settings. Despite its success in predicting patient outcomes in other clinical settings, few applications of machine learning to food allergy have been developed. Thus, in this study, we seek to leverage machine learning methodologies for OFC outcome prediction. Retrospective data was gathered from 1,112 patients who collectively underwent a total of 1,284 OFCs, and consisted of clinical factors including serum-specific Immunoglobulin E (IgE), total IgE, skin prick tests (SPTs), comorbidities, sex, and age. Using these features, multiple machine learning models were constructed to predict OFC outcomes for three common allergens: peanut, egg, and milk. The best performing model for each allergen was an ensemble of random forest (egg) or Learning Using Concave and Convex Kernels (LUCCK) (peanut, milk) models, which achieved an Area under the Curve (AUC) of 0.91, 0.96, and 0.94, in predicting OFC outcomes for peanut, egg, and milk, respectively. Moreover, all such models had sensitivity and specificity values 89%. Model interpretation via SHapley Additive exPlanations (SHAP) indicates that specific IgE, along with wheal and flare values from SPTs, are highly predictive of OFC outcomes. The results of this analysis suggest that ensemble learning has the potential to predict OFC outcomes and reveal relevant clinical factors for further study.

cs.LG↗

arXiv2021

A Novel Tropical Geometry-based Interpretable Machine Learning Method: Application in Prognosis of Advanced Heart Failure

A model's interpretability is essential to many practical applications such as clinical decision support systems. In this paper, a novel interpretable machine learning method is presented, which can model the relationship between input variables and responses in humanly understandable rules. The method is built by applying tropical geometry to fuzzy inference systems, wherein variable encoding functions and salient rules can be discovered by supervised learning. Experiments using synthetic datasets were conducted to investigate the performance and capacity of the proposed algorithm in classification and rule discovery. Furthermore, the proposed method was applied to a clinical application that identified heart failure patients that would benefit from advanced therapies such as heart transplant or durable mechanical circulatory support. Experimental results show that the proposed network achieved great performance on the classification tasks. In addition to learning humanly understandable rules from the dataset, existing fuzzy domain knowledge can be easily transferred into the network and used to facilitate model training. From our results, the proposed model and the ability of learning existing domain knowledge can significantly improve the model generalizability. The characteristics of the proposed network make it promising in applications requiring model reliability and justification.

cs.LG↗

arXiv2021

A Physics-Informed Neural Network Framework For Partial Differential Equations on 3D Surfaces: Time-Dependent Problems

In this paper, we show a physics-informed neural network solver for the time-dependent surface PDEs. Unlike the traditional numerical solver, no extension of PDE and mesh on the surface is needed. We show a simplified prior estimate of the surface differential operators so that PINN's loss value will be an indicator of the residue of the surface PDEs. Numerical experiments verify efficacy of our algorithm.

cs.LG↗

arXiv2020

Transparency Tools for Fairness in AI (Luskin)

We propose new tools for policy-makers to use when assessing and correcting fairness and bias in AI algorithms. The three tools are: - A new definition of fairness called "controlled fairness" with respect to choices of protected features and filters. The definition provides a simple test of fairness of an algorithm with respect to a dataset. This notion of fairness is suitable in cases where fairness is prioritized over accuracy, such as in cases where there is no "ground truth" data, only data labeled with past decisions (which may have been biased). - Algorithms for retraining a given classifier to achieve "controlled fairness" with respect to a choice of features and filters. Two algorithms are presented, implemented and tested. These algorithms require training two different models in two stages. We experiment with combinations of various types of models for the first and second stage and report on which combinations perform best in terms of fairness and accuracy. - Algorithms for adjusting model parameters to achieve a notion of fairness called "classification parity". This notion of fairness is suitable in cases where accuracy is prioritized. Two algorithms are presented, one which assumes that protected features are accessible to the model during testing, and one which assumes protected features are not accessible during testing. We evaluate our tools on three different publicly available datasets. We find that the tools are useful for understanding various dimensions of bias, and that in practice the algorithms are effective in starkly reducing a given observed bias when tested on new data.

cs.LG↗

arXiv2019

Contextual Attention for Hand Detection in the Wild

We present Hand-CNN, a novel convolutional network architecture for detecting hand masks and predicting hand orientations in unconstrained images. Hand-CNN extends MaskRCNN with a novel attention mechanism to incorporate contextual cues in the detection process. This attention mechanism can be implemented as an efficient network module that captures non-local dependencies between features. This network module can be inserted at different stages of an object detection network, and the entire detector can be trained end-to-end. We also introduce a large-scale annotated hand dataset containing hands in unconstrained images for training and evaluation. We show that Hand-CNN outperforms existing methods on several datasets, including our hand detection benchmark and the publicly available PASCAL VOC human layout challenge. We also conduct ablation studies on hand detection to show the effectiveness of the proposed contextual attention module.

cs.CV↗