On Expressivity of Height in Neural Networks

• URL: http://arxiv.org/abs/2305.07037v2
• Date: Sat, 04 Jan 2025 02:25:06 GMT
• Title: On Expressivity of Height in Neural Networks
• Authors: Feng-Lei Fan, Ze-Yu Li, Huan Xiong, Tieyong Zeng,
• Abstract summary: We call a neural network characterized by width, depth, and height a 3D network. We show via bound estimation and explicit construction that given the same number of neurons and parameters, a 3D ReLU network of width $W$, depth $K$, and height $H$ has greater expressive power than a 2D network of width $Htimes W$ and depth $K$.
• Score: 29.49793694185358
• License:
• Abstract: In this work, beyond width and depth, we augment a neural network with a new dimension called height by intra-linking neurons in the same layer to create an intra-layer hierarchy, which gives rise to the notion of height. We call a neural network characterized by width, depth, and height a 3D network. To put a 3D network in perspective, we theoretically and empirically investigate the expressivity of height. We show via bound estimation and explicit construction that given the same number of neurons and parameters, a 3D ReLU network of width $W$, depth $K$, and height $H$ has greater expressive power than a 2D network of width $H\times W$ and depth $K$, \textit{i.e.}, $\mathcal{O}((2^H-1)W)^K)$ vs $\mathcal{O}((HW)^K)$, in terms of generating more pieces in a piecewise linear function. Next, through approximation rate analysis, we show that by introducing intra-layer links into networks, a ReLU network of width $\mathcal{O}(W)$ and depth $\mathcal{O}(K)$ can approximate polynomials in $[0,1]^d$ with error $\mathcal{O}\left(2^{-2WK}\right)$, which improves $\mathcal{O}\left(W^{-K}\right)$ and $\mathcal{O}\left(2^{-K}\right)$ for fixed width networks. Lastly, numerical experiments on 5 synthetic datasets, 15 tabular datasets, and 3 image benchmarks verify that 3D networks can deliver competitive regression and classification performance.

Related papers

• Deep Neural Networks: Multi-Classification and Universal Approximation [0.0]
  We demonstrate that a ReLU deep neural network with a width of $2$ and a depth of $2N+4M-1$ layers can achieve finite sample memorization for any dataset comprising $N$ elements. We also provide depth estimates for approximating $W1,p$ functions and width estimates for approximating $Lp(Omega;mathbbRm)$ for $mgeq1$.
  arXiv  Detail & Related papers  (2024-09-10T14:31:21Z)
• Implicit Hypersurface Approximation Capacity in Deep ReLU Networks [0.0]
  We develop a geometric approximation theory for deep feed-forward neural networks with ReLU activations. We show that a deep fully-connected ReLU network of width $d+1$ can implicitly construct an approximation as its zero contour.
  arXiv  Detail & Related papers  (2024-07-04T11:34:42Z)
• Bayesian Inference with Deep Weakly Nonlinear Networks [57.95116787699412]
  We show at a physics level of rigor that Bayesian inference with a fully connected neural network is solvable. We provide techniques to compute the model evidence and posterior to arbitrary order in $1/N$ and at arbitrary temperature.
  arXiv  Detail & Related papers  (2024-05-26T17:08:04Z)
• Learning Hierarchical Polynomials with Three-Layer Neural Networks [56.71223169861528]
  We study the problem of learning hierarchical functions over the standard Gaussian distribution with three-layer neural networks. For a large subclass of degree $k$s $p$, a three-layer neural network trained via layerwise gradientp descent on the square loss learns the target $h$ up to vanishing test error. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.
  arXiv  Detail & Related papers  (2023-11-23T02:19:32Z)
• Rates of Approximation by ReLU Shallow Neural Networks [8.22379888383833]
  We show that ReLU shallow neural networks with $m$ hidden neurons can uniformly approximate functions from the H"older space. Such rates are very close to the optimal one $O(m-fracrd)$ in the sense that $fracd+2d+4d+4$ is close to $1$, when the dimension $d$ is large.
  arXiv  Detail & Related papers  (2023-07-24T00:16:50Z)
• Understanding Deep Neural Function Approximation in Reinforcement Learning via $\epsilon$-Greedy Exploration [53.90873926758026]
  This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) We focus on the value based algorithm with the $epsilon$-greedy exploration via deep (and two-layer) neural networks endowed by Besov (and Barron) function spaces. Our analysis reformulates the temporal difference error in an $L2(mathrmdmu)$-integrable space over a certain averaged measure $mu$, and transforms it to a generalization problem under the non-iid setting.
  arXiv  Detail & Related papers  (2022-09-15T15:42:47Z)
• Shallow neural network representation of polynomials [91.3755431537592]
  We show that $d$-variables of degreeR$ can be represented on $[0,1]d$ as shallow neural networks of width $d+1+sum_r=2Rbinomr+d-1d-1d-1[binomr+d-1d-1d-1[binomr+d-1d-1d-1[binomr+d-1d-1d-1d-1[binomr+d-1d-1d-1d-1
  arXiv  Detail & Related papers  (2022-08-17T08:14:52Z)
• Neural Network Architecture Beyond Width and Depth [4.468952886990851]
  This paper proposes a new neural network architecture by introducing an additional dimension called height beyond width and depth. It is shown that neural networks with three-dimensional architectures are significantly more expressive than the ones with two-dimensional architectures.
  arXiv  Detail & Related papers  (2022-05-19T10:29:11Z)
• Learning Over-Parametrized Two-Layer ReLU Neural Networks beyond NTK [58.5766737343951]
  We consider the dynamic of descent for learning a two-layer neural network. We show that an over-parametrized two-layer neural network can provably learn with gradient loss at most ground with Tangent samples.
  arXiv  Detail & Related papers  (2020-07-09T07:09:28Z)
• Deep Polynomial Neural Networks [77.70761658507507]
  $Pi$Nets are a new class of function approximators based on expansions. $Pi$Nets produce state-the-art results in three challenging tasks, i.e. image generation, face verification and 3D mesh representation learning.
  arXiv  Detail & Related papers  (2020-06-20T16:23:32Z)
• Sharp Representation Theorems for ReLU Networks with Precise Dependence on Depth [26.87238691716307]
  We prove sharp-free representation results for neural networks with $D$ ReLU layers under square loss. Our results confirm the prevailing hypothesis that deeper networks are better at representing less smooth functions.
  arXiv  Detail & Related papers  (2020-06-07T05:25:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.