Clustering in Causal Attention Masking

This work presents a modification of the self-attention dynamics proposed by Geshkovski et al. (arXiv:2312.10794) to better reflect the practically relevant, causally masked attention used in transformer architectures for generative AI. This modification translates into an interacting particle system that cannot be interpreted as a mean-field gradient flow. Despite this loss of structure, we significantly strengthen the results of Geshkovski et al. (arXiv:2312.10794) in this context: While previous rigorous results focused on cases where all three matrices (Key, Query, and Value) were scaled identities, we prove asymptotic convergence to a single cluster for arbitrary key-query matrices and a value matrix equal to the identity. Additionally, we establish a connection to the classical R\'enyi parking problem from combinatorial geometry to make initial theoretical steps towards demonstrating the existence of meta-stable states.

Optimal Quantization for Matrix Multiplication

Recent work in machine learning community proposed multiple methods for performing lossy compression (quantization) of large matrices. This quantization is important for accelerating matrix multiplication (main component of large language models), which is often bottlenecked by the speed of loading these matrices from memory. Unlike classical vector quantization and rate-distortion theory, the goal of these new compression algorithms is to be able to approximate not the matrices themselves, but their matrix product. Specifically, given a pair of real matrices $A,B$ an encoder (compressor) is applied to each of them independently producing descriptions with $R$ bits per entry. These representations subsequently are used by the decoder to estimate matrix product $A^\top B$. In this work, we provide a non-asymptotic lower bound on the mean squared error of this approximation (as a function of rate $R$) for the case of matrices $A,B$ with iid Gaussian entries. Algorithmically, we construct a universal quantizer based on nested lattices with an explicit guarantee of approximation error for any (non-random) pair of matrices $A$, $B$ in terms of only Frobenius norms $\|A\|_F, \|B\|_F$ and $\|A^\top B\|_F$. For iid Gaussian matrices our quantizer achieves the lower bound and is, thus, asymptotically optimal. A practical low-complexity version of our quantizer achieves performance quite close to optimal. In information-theoretic terms we derive rate-distortion function for matrix multiplication of iid Gaussian matrices.

Dynamic metastability in the self-attention model

We consider the self-attention model - an interacting particle system on the unit sphere, which serves as a toy model for Transformers, the deep neural network architecture behind the recent successes of large language models. We prove the appearance of dynamic metastability conjectured in [GLPR23] - although particles collapse to a single cluster in infinite time, they remain trapped near a configuration of several clusters for an exponentially long period of time. By leveraging a gradient flow interpretation of the system, we also connect our result to an overarching framework of slow motion of gradient flows proposed by Otto and Reznikoff [OR07] in the context of coarsening and the Allen-Cahn equation. We finally probe the dynamics beyond the exponentially long period of metastability, and illustrate that, under an appropriate time-rescaling, the energy reaches its global maximum in finite time and has a staircase profile, with trajectories manifesting saddle-to-saddle-like behavior, reminiscent of recent works in the analysis of training dynamics via gradient descent for two-layer neural networks.

RF Challenge: The Data-Driven Radio Frequency Signal Separation Challenge

This paper addresses the critical problem of interference rejection in radio-frequency (RF) signals using a novel, data-driven approach that leverages state-of-the-art AI models. Traditionally, interference rejection algorithms are manually tailored to specific types of interference. This work introduces a more scalable data-driven solution and contains the following contributions. First, we present an insightful signal model that serves as a foundation for developing and analyzing interference rejection algorithms. Second, we introduce the RF Challenge, a publicly available dataset featuring diverse RF signals along with code templates, which facilitates data-driven analysis of RF signal problems. Third, we propose novel AI-based rejection algorithms, specifically architectures like UNet and WaveNet, and evaluate their performance across eight different signal mixture types. These models demonstrate superior performance exceeding traditional methods like matched filtering and linear minimum mean square error estimation by up to two orders of magnitude in bit-error rate. Fourth, we summarize the results from an open competition hosted at 2024 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2024) based on the RF Challenge, highlighting the significant potential for continued advancements in this area. Our findings underscore the promise of deep learning algorithms in mitigating interference, offering a strong foundation for future research.

A mathematical perspective on Transformers

Transformers play a central role in the inner workings of large language models. We develop a mathematical framework for analyzing Transformers based on their interpretation as interacting particle systems, which reveals that clusters emerge in long time. Our study explores the underlying theory and offers new perspectives for mathematicians as well as computer scientists.

Kernel-Based Tests for Likelihood-Free Hypothesis Testing

Given $n$ observations from two balanced classes, consider the task of labeling an additional $m$ inputs that are known to all belong to \emph{one} of the two classes. Special cases of this problem are well-known: with complete knowledge of class distributions ($n=\infty$) the problem is solved optimally by the likelihood-ratio test; when $m=1$ it corresponds to binary classification; and when $m\approx n$ it is equivalent to two-sample testing. The intermediate settings occur in the field of likelihood-free inference, where labeled samples are obtained by running forward simulations and the unlabeled sample is collected experimentally. In recent work it was discovered that there is a fundamental trade-off between $m$ and $n$: increasing the data sample $m$ reduces the amount $n$ of training/simulation data needed. In this work we (a) introduce a generalization where unlabeled samples come from a mixture of the two classes -- a case often encountered in practice; (b) study the minimax sample complexity for non-parametric classes of densities under \textit{maximum mean discrepancy} (MMD) separation; and (c) investigate the empirical performance of kernels parameterized by neural networks on two tasks: detection of the Higgs boson and detection of planted DDPM generated images amidst CIFAR-10 images. For both problems we confirm the existence of the theoretically predicted asymmetric $m$ vs $n$ trade-off.

Score-based Source Separation with Applications to Digital Communication Signals

We propose a new method for separating superimposed sources using diffusion-based generative models. Our method relies only on separately trained statistical priors of independent sources to establish a new objective function guided by maximum a posteriori estimation with an $\alpha$-posterior, across multiple levels of Gaussian smoothing. Motivated by applications in radio-frequency (RF) systems, we are interested in sources with underlying discrete nature and the recovery of encoded bits from a signal of interest, as measured by the bit error rate (BER). Experimental results with RF mixtures demonstrate that our method results in a BER reduction of 95% over classical and existing learning-based methods. Our analysis demonstrates that our proposed method yields solutions that asymptotically approach the modes of an underlying discrete distribution. Furthermore, our method can be viewed as a multi-source extension to the recently proposed score distillation sampling scheme, shedding additional light on its use beyond conditional sampling.

The emergence of clusters in self-attention dynamics

Viewing Transformers as interacting particle systems, we describe the geometry of learned representations when the weights are not time dependent. We show that particles, representing tokens, tend to cluster toward particular limiting objects as time tends to infinity. Cluster locations are determined by the initial tokens, confirming context-awareness of representations learned by Transformers. Using techniques from dynamical systems and partial differential equations, we show that the type of limiting object that emerges depends on the spectrum of the value matrix. Additionally, in the one-dimensional case we prove that the self-attention matrix converges to a low-rank Boolean matrix. The combination of these results mathematically confirms the empirical observation made by Vaswani et al. [VSP'17] that leaders appear in a sequence of tokens when processed by Transformers.

On Neural Architectures for Deep Learning-based Source Separation of Co-Channel OFDM Signals

We study the single-channel source separation problem involving orthogonal frequency-division multiplexing (OFDM) signals, which are ubiquitous in many modern-day digital communication systems. Related efforts have been pursued in monaural source separation, where state-of-the-art neural architectures have been adopted to train an end-to-end separator for audio signals (as 1-dimensional time series). In this work, through a prototype problem based on the OFDM source model, we assess -- and question -- the efficacy of using audio-oriented neural architectures in separating signals based on features pertinent to communication waveforms. Perhaps surprisingly, we demonstrate that in some configurations, where perfect separation is theoretically attainable, these audio-oriented neural architectures perform poorly in separating co-channel OFDM waveforms. Yet, we propose critical domain-informed modifications to the network parameterization, based on insights from OFDM structures, that can confer about 30 dB improvement in performance.

The Sample Complexity of Approximate Rejection Sampling with Applications to Smoothed Online Learning

Suppose we are given access to $n$ independent samples from distribution $\mu$ and we wish to output one of them with the goal of making the output distributed as close as possible to a target distribution $\nu$. In this work we show that the optimal total variation distance as a function of $n$ is given by $\tilde\Theta(\frac{D}{f'(n)})$ over the class of all pairs $\nu,\mu$ with a bounded $f$-divergence $D_f(\nu\|\mu)\leq D$. Previously, this question was studied only for the case when the Radon-Nikodym derivative of $\nu$ with respect to $\mu$ is uniformly bounded. We then consider an application in the seemingly very different field of smoothed online learning, where we show that recent results on the minimax regret and the regret of oracle-efficient algorithms still hold even under relaxed constraints on the adversary (to have bounded $f$-divergence, as opposed to bounded Radon-Nikodym derivative). Finally, we also study efficacy of importance sampling for mean estimates uniform over a function class and compare importance sampling with rejection sampling.