Vinoth Nandakumar

Vinoth Nandakumar, Arush Tagade, Tongliang Liu

Over the past decade deep learning has revolutionized the field of computer vision, with convolutional neural network models proving to be very effective for image classification benchmarks. However, a fundamental theoretical questions remain answered: why can they solve discrete image classification tasks that involve feature extraction? We address this question in this paper by introducing a novel mathematical model for image classification, based on feature extraction, that can be used to generate images resembling real-world datasets. We show that convolutional neural network classifiers can solve these image classification tasks with zero error. In our proof, we construct piecewise linear functions that detect the presence of features, and show that they can be realized by a convolutional network.

Vinoth Nandakumar, Qiang Qu, Peng Mi et al.

Recent advancements in recurrent neural networks (RNNs) have reinvigorated interest in their application to natural language processing tasks, particularly with the development of more efficient and parallelizable variants known as state space models (SSMs), which have shown competitive performance against transformer models while maintaining a lower memory footprint. While RNNs and SSMs (e.g., Mamba) have been empirically more successful than rule-based systems based on n-gram models, a rigorous theoretical explanation for this success has not yet been developed, as it is unclear how these models encode the combinatorial rules that govern the next-word prediction task. In this paper, we construct state space language models that can solve the next-word prediction task for languages generated from n-gram rules, thereby showing that the former are more expressive. Our proof shows how SSMs can encode n-gram rules using new theoretical results on their memorization capacity, and demonstrates how their context window can be controlled by restricting the spectrum of the state transition matrix. We conduct experiments with a small dataset generated from n-gram rules to show how our framework can be applied to SSMs and RNNs obtained through gradient-based optimization.

Vinoth Nandakumar, Qiang Qu, Pramod Thebe et al.

Deep neural networks are widely believed to derive their expressive power from their ability to form \textbf{hierarchical representations}, capturing progressively more abstract and compositional features across layers. In language modeling, \textbf{transformers} have emerged as the dominant architecture, with early layers capturing local syntactic patterns and later layers encoding more complex clause-level dependencies. While this intuition has shaped model design, there remains a lack of rigorous theoretical work demonstrating \textbf{how} deep transformers represent such hierarchical structures. In this work, we analyze the expressiveness of deep transformer models through the formal lens of bounded-depth, non-recursive context-free grammars. For this class of grammars, we explicitly construct transformers with positional attention whose depth grows linearly with grammar depth, while the neuron count scales with the number of derivation-tree shapes and quadratically with the number of production rules. Our theoretical results support the linear representation hypothesis by demonstrating that these architectures possess the structural capacity to encode abstract grammatical states into low-dimensional, linearly separable subspaces within the residual stream.