Deep neural networks with piecewise-polynomial activations can approximate smooth functions and their derivatives.
problem Approximating smooth functions and their derivatives with neural networks.
method Derives the depth, width, and sparsity required for approximation in Hölder norms.
result Deep neural networks with bounded weights can approximate Hölder smooth functions and their derivatives.
Deep polynomial neural networks measure their expressiveness by the dimension of their functional space.
problem Measuring the expressiveness of deep polynomial neural networks.
method Analyzing the algebraic variety defined by the polynomial neural network's weights and activations.
result The dimension of the algebraic variety is a precise measure of the network's expressiveness.
Deep tensor factorization benefits from implicit regularization with polynomial growth.
problem Tensor factorization's implicit regularization effect in deep networks is not well understood.
method Investigated the implicit regularization in deep tensor factorization, showing polynomial growth.
result Implicit regularization in deep tensor factorization grows polynomially with depth, improving estimation accuracy and convergence.
Proposes a novel approach for deep neural network initialization using polynomial approximations.
problem Improving deep neural network training from initialization.
method Uses polynomial-based approximations to initialize deep neural networks.
result Networks initialized with polynomial approximations are more likely to achieve a desirable local minimum during training.
NN2Poly converts deep neural networks into polynomial models for better understanding.
problem Improving neural network interpretability and theoretical understanding.
method Taylor expansion on activation functions, combinatorial properties, and polynomial coefficients calculation.
result NN2Poly accurately represents deep feed-forward neural networks as polynomial models.
The paper proves deep neural networks with analytic activation can approximate any function.
problem Approximating functions with neural networks using analytic activation functions.
method Elementary proofs for real and complex networks, Stone-Weierstrass theorem, Mergelyan's theorem.
result Closure of neural network classes equals space of polynomials for analytic activation.
Deep networks better approximate functions with compositional structure.
problem Approximating functions with complex structures.
method Design deep networks with compositional structure, leveraging the blessing of compositionality.
result Deep networks can approximate functions better than shallow networks when the function has a compositional structure.
Study on identifiability of deep polynomial neural networks.
problem Understanding when polynomial neural networks can be uniquely identified.
method Comprehensive analysis including various architectures, using tensor decompositions and Kruskal-type theorems.
result Identifiability conditions for deep PNNs, including layer width and activation degree constraints.
Proposes Π-Nets, polynomial neural networks, for improved representation power.
problem Improving representation power in deep learning models.
method Introduces Π-Nets, a new class of deep polynomial neural networks. result Demonstrates Π-Nets outperform standard DCNNs and achieve state-of-the-art results. A new neural network model uses polynomial chaos theory to improve neural signal processing.
problem Redundant neural signal representation in DANNs.
method Employing arbitrary polynomial chaos theory to construct orthonormal representations in DANNs.
result Improves neural signal processing by reducing redundancy and enhancing orthogonality.
This paper presents a new mathematical framework to analyze the loss functions of deep neural networks with ReLU functions. Furthermore, as as application of this theory, we prove that the loss functions can reconstruct the inputs of the training samples up to scalar multiplication (as vectors) and can provide the numb…
Deep ReLU networks can approximate smooth functions nearly optimally.
problem Approximating smooth functions with deep neural networks.
method Using Taylor expansions and deep ReLU network approximations, the paper establishes optimal approximation error bounds.
result Deep ReLU networks of width and depth O(NlnN) and O(LlnL) can approximate f∈Cs([0,1]d) with an error O(∥f∥Cs([0,1]d)N−2s/dL−2s/d). An important conjecture in knot theory relates the large-N, double scaling limit of the colored Jones polynomial JK,N(q) of a knot K to the hyperbolic volume of the knot complement, Vol(K). A less studied question is whether Vol(K) can be recovered directly from the original Jones polynomial …
This work analyzes how different layers in deep neural networks contribute to generalization error.
problem Understanding the role of each layer in deep neural networks for generalization.
method Spectral analysis, Neural Tangent Kernel, Hermite polynomials, Spherical Harmonics.
result Initial layers in deep neural networks have a larger bias towards high-frequency functions.
Wide networks with polynomial activations have proven asymptotic behavior.
problem Understanding the behavior of neural networks in the large width limit.
method Proving a conjecture for deep networks with polynomial activation functions.
result Tight bounds on the behavior of wide networks during stochastic gradient descent and derivation of their finite-width dynamics.
Deep and wide networks are shown to be equivalent in terms of their capability.
problem The relationship between the width and depth of neural networks.
method Formulated transforms to map networks, used polynomial representations.
result Deep and wide networks are quasi-equivalent with an arbitrarily small error.
In this paper, we aim at introducing a new machine learning model, namely reconciled polynomial machine, which can provide a unified representation of existing shallow and deep machine learning models. Reconciled polynomial machine predicts the output by computing the inner product of the feature kernel function and va…
Algorithm efficiently learns deep ReLU networks with polynomial runtime in depth and parameters.
problem Learning deep ReLU networks with polynomial runtime.
method Algorithm using filtered PCA and lattice polynomial analysis.
result First nontrivial results for networks of depth more than two with polynomial runtime.
Proposes polynomial neural networks for improved function approximation in various tasks.
problem Improving function approximation in various tasks like image generation, face verification, and 3D mesh representation learning.
method Introduces polynomial neural networks (Π-Nets) and three tensor decompositions to reduce parameter count and enhance expressiveness. result Demonstrates that Π-Nets can produce state-of-the-art results in challenging tasks without non-linear activation functions. We consider deep neural networks, in which the output of each node is a quadratic function of its inputs. Similar to other deep architectures, these networks can compactly represent any function on a finite training set. The main goal of this paper is the derivation of an efficient layer-by-layer algorithm for training…
Path regularization reveals convex optimization in deep ReLU networks.
problem Understanding the optimization landscape of deep neural networks.
method Introducing path regularization to make the training problem convex and sparsity-inducing.
result Path regularized parallel ReLU networks are a parsimonious convex model in high dimensions.
Polynomial neural networks explore thresholds for maximum expressiveness.
problem Understanding the limits of polynomial neural networks' expressiveness.
method Introducing activation degree threshold to measure network expressiveness and proving its existence and upper bounds.
result Polynomial neural networks with equi-width architectures achieve the maximum expressiveness.
PolyLUT uses polynomials to reduce FPGA latency.
problem Reducing latency in FPGA-based neural network inference.
method Training neural networks using multivariate polynomials as basic building blocks.
result Achieved significant latency and area improvements.
RPN unifies various models with a reconciled polynomial network.
problem Unifying diverse models for deep function learning.
method RPN disentangles functions into inner products of expansion and reconciliation functions.
result RPN accurately approximates underlying functions for data distributions.
Shallow nonlinear networks can separate classes linearly with polynomially scaling width.
problem Understanding the linear separability of deep networks' features.
method Modeling inputs as a union of low-dimensional subspaces and using random weights and quadratic activations.
result Shallow nonlinear networks can achieve linear separation with polynomially scaling width.
New theory predicts deep neural network learning curves.
problem Understanding and optimizing deep neural networks.
method Gaussian field theory and renormalization group.
result Accurate predictions of deep neural network learning curves.
Study robust algorithms for deep networks under adversarial perturbations.
problem Designing robust deep networks to adversarial attacks.
method Connecting robustness to polynomial optimization, designing efficient algorithms.
result Provable robustness for linear classifiers and PTFs, characterizing robustness price, efficient certification.
The paper analyzes deep neural networks' expressivity and training, revealing critical expressivity issues.
problem Critical expressivity issues in deep neural networks.
method Quantitative analysis using Hilbert space and Hermite polynomials for feature mapping and activation function design.
result Deep neural networks evolve to the edge of chaos, but expressivity depends on overcoming convergence.
POUnets combine partitions of unity and monomials for efficient deep learning.
problem Efficiently approximating functions with deep neural networks in high dimensions.
method Integrates partitions of unity and monomials into neural network architecture.
result POUnets achieve hp-convergence for smooth functions and outperform MLPs for discontinuous functions.
The paper proves barriers to approximating functions with small weights and depth in neural networks.
problem Proving barriers to approximating functions with constant depth neural networks.
method Reduction to open problems and natural-proof barriers in circuit complexity, and a new approach to polynomially-bounded functions.
result There are fundamental barriers to proving results beyond depth 4 for constant-depth neural networks.
The problem of explaining the behavior of deep neural networks has recently gained a lot of attention. While several attribution methods have been proposed, most come without strong theoretical foundations, which raises questions about their reliability. On the other hand, the literature on cooperative game theory sugg…
Improved pseudo-label accuracy in semi-supervised learning with Hermite polynomials.
problem Improving pseudo-label accuracy in semi-supervised learning.
method Substituting ReLU activations with Hermite polynomial activations in deep networks.
result Hermite polynomial activations yield significant improvements in pseudo-label accuracy and financial savings.
Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely …
Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. The current paper proves gradient descent achieves zero training loss in polynomial time for a deep over-parameterized neural network with residual connections (ResNet). Our analysis relies on the p…
The remarkable development of deep learning in medicine and healthcare domain presents obvious privacy issues, when deep neural networks are built on users' personal and highly sensitive data, e.g., clinical records, user profiles, biomedical images, etc. However, only a few scientific studies on preserving privacy in …
The staircase property aids deep learning by guiding hierarchical feature learning.
problem Understanding how hierarchical structure influences deep learning performance.
method Defined and proved the staircase property for Boolean hypercube functions, and showed its learnability by layerwise stochastic coordinate descent.
result Staircase functions can be learned in polynomial time using layerwise stochastic coordinate descent on regular neural networks.
As the success of deep learning reaches more grounds, one would like to also envision the potential limits of deep learning. This paper gives a first set of results proving that certain deep learning algorithms fail at learning certain efficiently learnable functions. The results put forward a notion of cross-predictab…
Three-layer networks learn complex hierarchical polynomials of multiple nonlinear features.
problem Understanding how neural networks learn hierarchical features of multiple nonlinear inputs.
method Examine a broad class of functions using three-layer neural networks, showing complete recovery and efficient learning.
result Three-layer neural networks trained via gradient descent can learn hierarchical polynomials of multiple nonlinear features efficiently.
nn2poly converts neural networks into interpretable polynomial models.
problem Interpreting complex neural networks.
method NN2Poly method for converting neural networks into polynomial models.
result Captures variable interactions and provides interpretable coefficients.
Paper develops exact convex optimization for neural networks with polynomial activations.
problem Training two-layer neural networks with nonlinear polynomial activations.
method Exact convex optimization using semidefinite programming.
result Global optimization of neural networks is polynomial-time computable.
Polynomial Chaos Expansion improves operator learning for PDEs.
problem Approximating mappings between infinite-dimensional functional spaces.
method Polynomial Chaos Expansion (PCE) for operator learning.
result PCE achieves strong performance in operator learning and uncertainty quantification.
Deep neural networks predict knot invariants across dimensions with high accuracy.
problem Predicting knot invariants in different dimensions using machine learning.
method Two-layer feed-forward neural networks trained on various knot invariants.
result Neural networks achieve high accuracy in predicting knot invariants like s and g. Great successes of deep neural networks have been witnessed in various real applications. Many algorithmic and implementation techniques have been developed, however, theoretical understanding of many aspects of deep neural networks is far from clear. A particular interesting issue is the usefulness of dropout, which w…
Gradient descent converges globally in deep linear residual networks with ZAS initialization.
problem Optimizing deep linear residual networks for convergence.
method Zero-asymmetric (ZAS) initialization for gradient descent.
result Gradient descent converges to an ε-optimal point in O(L^3 log(1/ε)) iterations.
The paper shows how neural networks can approximate PDEs with polynomial scaling in dimension.
problem Understanding the complexity of approximating PDE solutions with neural networks.
method Developed a proof technique to simulate gradient descent using neural networks.
result Neural network parameters scale polynomially with input dimension for approximating PDE solutions.
Gradient descent methods for deep ReLU networks achieve optimal generalization rates.
problem Generalization of gradient descent methods for deep neural networks
method Establishing minimax-optimal rates for GD and SGD with deep ReLU networks
result Gradient descent methods for deep ReLU networks achieve optimal generalization rates
Paper proposes DPN for encrypted speech recognition, maintaining privacy and security.
problem Privacy and security issues in cloud-based speech recognition.
method Deep Polynomial Network (DPN) for encrypted speech, cloud-local joint decoding.
result DPN can make frame-level predictions over encrypted speech and return them in encrypted form.
We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (Bölcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimensional oscillatory textures, and the Weierstrass function, a fractal function which is continuous but nowhere differentiable. Together with thei…