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Ahmed Roman

Publications and source records attributed to Ahmed Roman.

15 recordsLinked to original sources
arXiv2025

HARDMath2: A Benchmark for Applied Mathematics Built by Students as Part of a Graduate Class

Large language models (LLMs) have shown remarkable progress in mathematical problem-solving, but evaluation has largely focused on problems that have exact analytical solutions or involve formal proofs, often overlooking approximation-based problems ubiquitous in applied science and engineering. To fill this gap, we build on prior work and present HARDMath2, a dataset of 211 original problems covering the core topics in an introductory graduate applied math class, including boundary-layer analysis, WKB methods, asymptotic solutions of nonlinear partial differential equations, and the asymptotics of oscillatory integrals. This dataset was designed and verified by the students and instructors of a core graduate applied mathematics course at Harvard. We build the dataset through a novel collaborative environment that challenges students to write and refine difficult problems consistent with the class syllabus, peer-validate solutions, test different models, and automatically check LLM-generated solutions against their own answers and numerical ground truths. Evaluation results show that leading frontier models still struggle with many of the problems in the dataset, highlighting a gap in the mathematical reasoning skills of current LLMs. Importantly, students identified strategies to create increasingly difficult problems by interacting with the models and exploiting common failure modes. This back-and-forth with the models not only resulted in a richer and more challenging benchmark but also led to qualitative improvements in the students' understanding of the course material, which is increasingly important as we enter an age where state-of-the-art language models can solve many challenging problems across a wide domain of fields.

cs.LG↗
arXiv2024

LayerSkip: Enabling Early Exit Inference and Self-Speculative Decoding

We present LayerSkip, an end-to-end solution to speed-up inference of large language models (LLMs). First, during training we apply layer dropout, with low dropout rates for earlier layers and higher dropout rates for later layers, and an early exit loss where all transformer layers share the same exit. Second, during inference, we show that this training recipe increases the accuracy of early exit at earlier layers, without adding any auxiliary layers or modules to the model. Third, we present a novel self-speculative decoding solution where we exit at early layers and verify and correct with remaining layers of the model. Our proposed self-speculative decoding approach has less memory footprint than other speculative decoding approaches and benefits from shared compute and activations of the draft and verification stages. We run experiments on different Llama model sizes on different types of training: pretraining from scratch, continual pretraining, finetuning on specific data domain, and finetuning on specific task. We implement our inference solution and show speedups of up to 2.16x on summarization for CNN/DM documents, 1.82x on coding, and 2.0x on TOPv2 semantic parsing task. We open source our code and checkpoints at https://github.com/facebookresearch/LayerSkip.

cs.CL↗
arXiv2023

Simultaneous Dimensionality Reduction: A Data Efficient Approach for Multimodal Representations Learning

We explore two primary classes of approaches to dimensionality reduction (DR): Independent Dimensionality Reduction (IDR) and Simultaneous Dimensionality Reduction (SDR). In IDR methods, of which Principal Components Analysis is a paradigmatic example, each modality is compressed independently, striving to retain as much variation within each modality as possible. In contrast, in SDR, one simultaneously compresses the modalities to maximize the covariation between the reduced descriptions while paying less attention to how much individual variation is preserved. Paradigmatic examples include Partial Least Squares and Canonical Correlations Analysis. Even though these DR methods are a staple of statistics, their relative accuracy and data set size requirements are poorly understood. We introduce a generative linear model to synthesize multimodal data with known variance and covariance structures to examine these questions. We assess the accuracy of the reconstruction of the covariance structures as a function of the number of samples, signal-to-noise ratio, and the number of varying and covarying signals in the data. Using numerical experiments, we demonstrate that linear SDR methods consistently outperform linear IDR methods and yield higher-quality, more succinct reduced-dimensional representations with smaller datasets. Remarkably, regularized CCA can identify low-dimensional weak covarying structures even when the number of samples is much smaller than the dimensionality of the data, which is a regime challenging for all dimensionality reduction methods. Our work corroborates and explains previous observations in the literature that SDR can be more effective in detecting covariation patterns in data. These findings suggest that SDR should be preferred to IDR in real-world data analysis when detecting covariation is more important than preserving variation.

stat.ML↗
arXiv2022

Measuring gravitational force from Femto-gram source masses

Gravity is the weakest of all known forces. Measuring the force of gravity from micro and nano-scale source masses is an essential first step toward low-energy quantum gravity tests. In addition, measuring gravitational forces where the center-of-mass inter-distance is at the sub-mm scale extends the experimentally achievable parameter space for tests of Yukawa-like corrections to Newtonian gravity and tests for higher dimensions proposed to resolve the hierarchy problem of fundamental forces. Here, we propose an experiment using two optically trapped particles in ultrahigh vacuum conditions where the center of mass inter-distance is on the order of $10^2 nm$. In the proposed experiment, the source mass is a rotating Janus nano-particle such that the test mass (sensor) experiences a periodic gravitational potential. Using realistic experimental parameters, a signal-to-noise ratio $\geq 1$ is obtained for a Janus particle with radius $\geq 10^2 nm$ and a mass $\geq \text{10 } fg$. The proposed experiment extends the search of Yukawa corrections to gravity at $\approx 10^{-5}$ times gravity regime at $10^{2}nm $ interaction range, opens the door to low energy tests for quantum gravity, and enables direct experimental tests of extra-dimensional solutions to the hierarchy problem.

gr-qc↗
arXiv2022

Low probability states, data statistics, and entropy estimation

A fundamental problem in analysis of complex systems is getting a reliable estimate of entropy of their probability distributions over the state space. This is difficult because unsampled states can contribute substantially to the entropy, while they do not contribute to the Maximum Likelihood estimator of entropy, which replaces probabilities by the observed frequencies. Bayesian estimators overcome this obstacle by introducing a model of the low-probability tail of the probability distribution. Which statistical features of the observed data determine the model of the tail, and hence the output of such estimators, remains unclear. Here we show that well-known entropy estimators for probability distributions on discrete state spaces model the structure of the low probability tail based largely on few statistics of the data: the sample size, the Maximum Likelihood estimate, the number of coincidences among the samples, the dispersion of the coincidences. We derive approximate analytical entropy estimators for undersampled distributions based on these statistics, and we use the results to propose an intuitive understanding of how the Bayesian entropy estimators work.

physics.data-an↗
arXiv2022

Multi-dimensional structure of C. elegans thermal learning

Quantitative models of associative learning that explain the behavior of real animals with high precision have turned out very difficult to construct. We do this in the context of the dynamics of the thermal preference of C. elegans. For this, we quantify C. elegans thermotaxis in response to various conditioning parameters, genetic perturbations, and operant behavior using a fast, high-throughput microfluidic droplet assay. We then model this data comprehensively, within a new, biologically interpretable, multi-modal framework. We discover that the dynamics of thermal preference are described by two independent contributions and require a model with at least four dynamical variables. One pathway positively associates the experienced temperature independently of food and the other negatively associates to the temperature when food is absent.

physics.bio-ph↗
arXiv2022

Ballistic deposition with memory: a new universality class of surface growth with a new scaling law

Motivated by recent experimental studies in microbiology, we suggest a modification of the classic ballistic deposition model of surface growth, where the memory of a deposition at a site induces more depositions at that site or its neighbors. By studying the statistics of surfaces in this model, we obtain three independent critical exponents: the growth exponent $β=5/4$, the roughening exponent $α= 2$, and the new (size) exponent $γ= 1/2$. The model requires a modification to the Family-Vicsek scaling, resulting in the dynamical exponent $z = \frac{α+γ}β = 2$. This modified scaling collapses the surface width vs time curves for various lattice sizes. This is a previously unobserved universality class of surface growth that could describe surface properties of a wide range of natural systems.

nlin.CG↗
arXiv2017

Linear Dependency of Translations and Square Integrable Representations

Let $G$ be a locally compact group. We examine the problem of determining when nonzero functions in $L^2(G)$ have linearly independent translations. In particular, we establish some results for the case when $G$ has an irreducible, square integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group and the Weyl-Heisenberg group. We also investigate the case when $G$ has an abelian, closed subgroup of finite index.

math.FA↗
arXiv2016

A theoretical approach to understand spatial organization in complex ecologies

Predicting the fate of ecologies is a daunting, albeit extremely important, task. As part of this task one needs to develop an understanding of the organization, hierarchies, and correlations among the species forming the ecology. Focusing on complex food networks we present a theoretical method that allows to achieve this understanding. Starting from the adjacency matrix the method derives specific matrices that encode the various inter-species relationships. The full potential of the method is achieved in a spatial setting where one obtains detailed predictions for the emerging space-time patterns. For a variety of cases these theoretical predictions are verified through numerical simulations.

q-bio.PE↗
arXiv2016

A New Family of Bounded Divergence Measures and Application to Signal Detection

We introduce a new one-parameter family of divergence measures, called bounded Bhattacharyya distance (BBD) measures, for quantifying the dissimilarity between probability distributions. These measures are bounded, symmetric and positive semi-definite and do not require absolute continuity. In the asymptotic limit, BBD measure approaches the squared Hellinger distance. A generalized BBD measure for multiple distributions is also introduced. We prove an extension of a theorem of Bradt and Karlin for BBD relating Bayes error probability and divergence ranking. We show that BBD belongs to the class of generalized Csiszar f-divergence and derive some properties such as curvature and relation to Fisher Information. For distributions with vector valued parameters, the curvature matrix is related to the Fisher-Rao metric. We derive certain inequalities between BBD and well known measures such as Hellinger and Jensen-Shannon divergence. We also derive bounds on the Bayesian error probability. We give an application of these measures to the problem of signal detection where we compare two monochromatic signals buried in white noise and differing in frequency and amplitude.

math.ST↗
arXiv2015

Position and momentum uncertainties of a particle in a V-shaped potential under the minimal length uncertainty relation

We calculate the uncertainties in the position and momentum of a particle in the 1D potential V(x)=F|x|, F>0, when the position and momentum operators obey the deformed commutation relation [x,p]=i\hbar(1+βp^2), β>0. As in the harmonic oscillator case, which was investigated in a previous publication, the Hamiltonian H_1 = p^2/2m + F|x| admits discrete positive energy eigenstates for both positive and negative mass. The uncertainties for the positive mass states behave as Δx ~ 1/Δp as in the β=0 limit. For the negative mass states, however, in contrast to the harmonic oscillator case where we had Δx ~ Δp, both Δx and Δp diverge. We argue that the existence of the negative mass states and the divergence of their uncertainties can be understood by taking the classical limit of the theory. Comparison of our results is made with previous work by Benczik.

hep-th↗
arXiv2014

Spirals and coarsening patterns in the competition of many species: A complex Ginzburg-Landau approach

In order to model real ecological systems one has to consider many species that interact in complex ways. However, most of the recent theoretical studies have been restricted to few species systems with rather trivial interactions. The few studies dealing with larger number of species and/or more complex interaction schemes are mostly restricted to numerical explorations. In this paper we determine, starting from the deterministic mean-field rate equations, for large classes of systems the space of coexistence fixed points at which biodiversity is maximal. For systems with a single coexistence fixed point we derive complex Ginzburg-Landau equations that allow to describe space-time pattern realized in two space dimensions. For selected cases we compare the theoretical predictions with the pattern observed in numerical simulations.

q-bio.PE↗
arXiv2013

Interplay between partnership formation and competition in generalized May-Leonard games

In order to better understand the interplay of partnership and competition in population dynamics we study a family of generalized May-Leonard models with $N$ species. These models have a very rich structure, characterized by different types of space-time patterns. Interesting partnership formations emerge following the maxim that "the enemy of my enemy is my friend." In specific cases cyclic dominance within coarsening clusters yields a peculiar coarsening behavior with intriguing pattern formation. We classify the different types of dynamics through the analysis of the square of the adjacency matrix. The dependence of the population densities on emerging pattern and propagating wave fronts is elucidated through a Fourier analysis. Finally, after having identified collaborating teams, we study interface fluctuations where we initially populate different parts of the system with different teams.

cond-mat.stat-mech↗
arXiv2012

Cyclic competition of four species: domains and interfaces

We study numerically domain growth and interface fluctuations in one- and two-dimensional lattice systems composed of four species that interact in a cyclic way. Particle mobility is implemented through exchanges of particles located on neighboring lattice sites. For the chain we find that domain growth strongly depends on the mobility, with a higher mobility yielding a larger domain growth exponent. In two space dimensions, when also exchanges between mutually neutral particles are possible, both domain growth and interface fluctuations display universal regimes that are independent of the predation and exchange rates.

cond-mat.stat-mech↗