Constructive Universal Approximation Theorems for Deep Joint-Equivariant Networks by Schur's Lemma

We present a unified constructive universal approximation theorem covering a wide range of learning machines including both shallow and deep neural networks based on the group representation theory. Constructive here means that the distribution of parameters is given in a closed-form expression (called the ridgelet transform). Contrary to the case of shallow models, expressive power analysis of deep models has been conducted in a case-by-case manner. Recently, Sonoda et al. (2023a,b) developed a systematic method to show a constructive approximation theorem from scalar-valued joint-group-invariant feature maps, covering a formal deep network. However, each hidden layer was formalized as an abstract group action, so it was not possible to cover real deep networks defined by composites of nonlinear activation function. In this study, we extend the method for vector-valued joint-group-equivariant feature maps, so to cover such real networks.

A unified Fourier slice method to derive ridgelet transform for a variety of depth-2 neural networks

To investigate neural network parameters, it is easier to study the distribution of parameters than to study the parameters in each neuron. The ridgelet transform is a pseudo-inverse operator that maps a given function $f$ to the parameter distribution $\gamma$ so that a network $\mathtt{NN}[\gamma]$ reproduces $f$, i.e. $\mathtt{NN}[\gamma]=f$. For depth-2 fully-connected networks on a Euclidean space, the ridgelet transform has been discovered up to the closed-form expression, thus we could describe how the parameters are distributed. However, for a variety of modern neural network architectures, the closed-form expression has not been known. In this paper, we explain a systematic method using Fourier expressions to derive ridgelet transforms for a variety of modern networks such as networks on finite fields $\mathbb{F}_p$, group convolutional networks on abstract Hilbert space $\mathcal{H}$, fully-connected networks on noncompact symmetric spaces $G/K$, and pooling layers, or the $d$-plane ridgelet transform.

A Policy Gradient Primal-Dual Algorithm for Constrained MDPs with Uniform PAC Guarantees

We study a primal-dual reinforcement learning (RL) algorithm for the online constrained Markov decision processes (CMDP) problem, wherein the agent explores an optimal policy that maximizes return while satisfying constraints. Despite its widespread practical use, the existing theoretical literature on primal-dual RL algorithms for this problem only provides sublinear regret guarantees and fails to ensure convergence to optimal policies. In this paper, we introduce a novel policy gradient primal-dual algorithm with uniform probably approximate correctness (Uniform-PAC) guarantees, simultaneously ensuring convergence to optimal policies, sublinear regret, and polynomial sample complexity for any target accuracy. Notably, this represents the first Uniform-PAC algorithm for the online CMDP problem. In addition to the theoretical guarantees, we empirically demonstrate in a simple CMDP that our algorithm converges to optimal policies, while an existing algorithm exhibits oscillatory performance and constraint violation.

Joint Group Invariant Functions on Data-Parameter Domain Induce Universal Neural Networks

The symmetry and geometry of input data are considered to be encoded in the internal data representation inside the neural network, but the specific encoding rule has been less investigated. By focusing on a joint group invariant function on the data-parameter domain, we present a systematic rule to find a dual group action on the parameter domain from a group action on the data domain. Further, we introduce generalized neural networks induced from the joint invariant functions, and present a new group theoretic proof of their universality theorems by using Schur's lemma. Since traditional universality theorems were demonstrated based on functional analytical methods, this study sheds light on the group theoretic aspect of the approximation theory, connecting geometric deep learning to abstract harmonic analysis.

Deep Ridgelet Transform: Voice with Koopman Operator Proves Universality of Formal Deep Networks

We identify hidden layers inside a DNN with group actions on the data space, and formulate the DNN as a dual voice transform with respect to Koopman operator, a linear representation of the group action. Based on the group theoretic arguments, particularly by using Schur's lemma, we show a simple proof of the universality of those DNNs.

LPML: LLM-Prompting Markup Language for Mathematical Reasoning

In utilizing large language models (LLMs) for mathematical reasoning, addressing the errors in the reasoning and calculation present in the generated text by LLMs is a crucial challenge. In this paper, we propose a novel framework that integrates the Chain-of-Thought (CoT) method with an external tool (Python REPL). We discovered that by prompting LLMs to generate structured text in XML-like markup language, we could seamlessly integrate CoT and the external tool and control the undesired behaviors of LLMs. With our approach, LLMs can utilize Python computation to rectify errors within CoT. We applied our method to ChatGPT (GPT-3.5) to solve challenging mathematical problems and demonstrated that combining CoT and Python REPL through the markup language enhances the reasoning capability of LLMs. Our approach enables LLMs to write the markup language and perform advanced mathematical reasoning using only zero-shot prompting.

Koopman-Based Bound for Generalization: New Aspect of Neural Networks Regarding Nonlinear Noise Filtering

We propose a new bound for generalization of neural networks using Koopman operators. Unlike most of the existing works, we focus on the role of the final nonlinear transformation of the networks. Our bound is described by the reciprocal of the determinant of the weight matrices and is tighter than existing norm-based bounds when the weight matrices do not have small singular values. According to existing theories about the low-rankness of the weight matrices, it may be counter-intuitive that we focus on the case where singular values of weight matrices are not small. However, motivated by the final nonlinear transformation, we can see that our result sheds light on a new perspective regarding a noise filtering property of neural networks. Since our bound comes from Koopman operators, this work also provides a connection between operator-theoretic analysis and generalization of neural networks. Numerical results support the validity of our theoretical results.

Quantum Ridgelet Transform: Winning Lottery Ticket of Neural Networks with Quantum Computation

Ridgelet transform has been a fundamental mathematical tool in the theoretical studies of neural networks. However, the practical applicability of ridgelet transform to conducting learning tasks was limited since its numerical implementation by conventional classical computation requires an exponential runtime $\exp(O(D))$ as data dimension $D$ increases. To address this problem, we develop a quantum ridgelet transform (QRT), which implements the ridgelet transform of a quantum state within a linear runtime $O(D)$ of quantum computation. As an application, we also show that one can use QRT as a fundamental subroutine for quantum machine learning (QML) to efficiently find a sparse trainable subnetwork of large shallow wide neural networks without conducting large-scale optimization of the original network. This application discovers an efficient way in this regime to demonstrate the lottery ticket hypothesis on finding such a sparse trainable neural network. These results open an avenue of QML for accelerating learning tasks with commonly used classical neural networks.

Universality of group convolutional neural networks based on ridgelet analysis on groups

We investigate the approximation property of group convolutional neural networks (GCNNs) based on the ridgelet theory. We regard a group convolution as a matrix element of a group representation, and formulate a versatile GCNN as a nonlinear mapping between group representations, which covers typical GCNN literatures such as a cyclic convolution on a multi-channel image, permutation-invariant datasets (Deep Sets), and $\mathrm{E}(n)$-equivariant convolutions. The ridgelet transform is an analysis operator of a depth-2 network, namely, it maps an arbitrary given target function $f$ to the weight $\gamma$ of a network $S[\gamma]$ so that the network represents the function as $S[\gamma]=f$. It has been known only for fully-connected networks, and this study is the first to present the ridgelet transform for (G)CNNs. Since the ridgelet transform is given as a closed-form integral operator, it provides a constructive proof of the $cc$-universality of GCNNs. Unlike previous universality arguments on CNNs, we do not need to convert/modify the networks into other universal approximators such as invariant polynomials and fully-connected networks.

Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysis

Neural network on Riemannian symmetric space such as hyperbolic space and the manifold of symmetric positive definite (SPD) matrices is an emerging subject of research in geometric deep learning. Based on the well-established framework of the Helgason-Fourier transform on the noncompact symmetric space, we present a fully-connected network and its associated ridgelet transform on the noncompact symmetric space, covering the hyperbolic neural network (HNN) and the SPDNet as special cases. The ridgelet transform is an analysis operator of a depth-2 continuous network spanned by neurons, namely, it maps an arbitrary given function to the weights of a network. Thanks to the coordinate-free reformulation, the role of nonlinear activation functions is revealed to be a wavelet function, and the reconstruction formula directly yields the universality of the proposed networks.