arXiv research

This paper improves deep neural network approximation for fully connected networks, achieving optimal convergence rates.

Deep RNNs compute American option prices and deltas efficiently.

In this paper we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approxima…

In this study, we propose a novel deep neural network and its supervised learning method that uses a feedforward supervisory signal. The method is inspired by the human visual system and performs human-like association-based learning without any backward error propagation. The feedforward supervisory signal that produc…

We provide a characterization of two types of directed homology for fully-connected, feedforward neural network architectures. These exact characterizations of the directed homology structure of a neural network architecture are the first of their kind. We show that the directed flag homology of deep networks reduces t…

We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden laye…

Residual neural networks don't help overcome sampling complexity issues.

Deep residual networks (ResNets) have demonstrated better generalization performance than deep feedforward networks (FFNets). However, the theory behind such a phenomenon is still largely unknown. This paper studies this fundamental problem in deep learning from a so-called "neural tangent kernel" perspective. Specific…

Review of neural network expressivity and architectures.

Predicts trainability of deep neural networks using reconstruction entropy.

We study deep neural networks and their use in semiparametric inference. We establish novel rates of convergence for deep feedforward neural nets. Our new rates are sufficiently fast (in some cases minimax optimal) to allow us to establish valid second-step inference after first-step estimation with deep learning, a re…

Conditions for uniquely identifying parameters of deep ReLU networks.

How to model and encode the semantics of human-written text and select the type of neural network to process it are not settled issues in sentiment analysis. Accuracy and transferability are critical issues in machine learning in general. These properties are closely related to the loss estimates for the trained model.…

In this paper, we present a new approach to interpret deep learning models. By coupling mutual information with network science, we explore how information flows through feedforward networks. We show that efficiently approximating mutual information allows us to create an information measure that quantifies how much in…

Study on deep neural networks using concentration inequalities and optimal stopping.

Time series anomaly detection is usually formulated as finding outlier data points relative to some usual data, which is also an important problem in industry and academia. To ensure systems working stably, internet companies, banks and other companies need to monitor time series, which is called KPI (Key Performance I…

This paper presents the application of a newly developed nature-inspired metaheuristic optimization method, namely the Adaptive Wind Driven Optimization (AWDO), to the training of feedforward artificial neural networks (NN) and presents a discussion into the future research of AWDO implementation in Deep Learning (DL).…

TVS-FNNs can approximate any continuous function on expanded input spaces.

Recent years, many researches attempt to open the black box of deep neural networks and propose a various of theories to understand it. Among them, Information Bottleneck (IB) theory claims that there are two distinct phases consisting of fitting phase and compression phase in the course of training. This statement att…

We simplify neural networks to 3D to study their topological changes.

In this chapter we take a look at the universal approximation question for stochastic feedforward neural networks. In contrast to deterministic networks, which represent mappings from a set of inputs to a set of outputs, stochastic networks represent mappings from a set of inputs to a set of probability distributions o…

This note presents in a technical though hopefully pedagogical way the three most common forms of neural network architectures: Feedforward, Convolutional and Recurrent. For each network, their fundamental building blocks are detailed. The forward pass and the update rules for the backpropagation algorithm are then der…

Implicit deep learning prediction rules generalize the recursive rules of feedforward neural networks. Such rules are based on the solution of a fixed-point equation involving a single vector of hidden features, which is thus only implicitly defined. The implicit framework greatly simplifies the notation of deep learni…

Neural networks can approximate functions uniformly across various measures.

The prediction of stock prices is an important task in economics, investment and making financial decisions. This has, for decades, spurred the interest of many researchers to make focused contributions to the design of accurate stock price predictive models; of which some have been utilized to predict the next day ope…

Study uses neural networks to improve option pricing accuracy.

This paper proposes a novel approach to train deep neural networks by unlocking the layer-wise dependency of backpropagation training. The approach employs additional modules called local critic networks besides the main network model to be trained, which are used to obtain error gradients without complete feedforward …

Neuromorphic hardware tends to pose limits on the connectivity of deep networks that one can run on them. But also generic hardware and software implementations of deep learning run more efficiently for sparse networks. Several methods exist for pruning connections of a neural network after it was trained without conne…

We establish a margin based data dependent generalization error bound for a general family of deep neural networks in terms of the depth and width, as well as the Jacobian of the networks. Through introducing a new characterization of the Lipschitz properties of neural network family, we achieve significantly tighter g…

In real world scenarios, objects are often partially occluded. This requires a robustness for object recognition against these perturbations. Convolutional networks have shown good performances in classification tasks. The learned convolutional filters seem similar to receptive fields of simple cells found in the prima…

LCW reduces activation shift in neural networks, improving training efficiency and generalization.

This paper examines properties of feedforward graphs to improve neural network performance.

Neural networks' feature geometry evolves like discrete Ricci flow.

Backdoors in deep neural networks are undetectable and enable invariance-based adversarial examples.

New insights into neural network training efficiency.

Math theory explains how neural networks learn abstract representations.

Gene expression data represents a unique challenge in predictive model building, because of the small number of samples $(n)$ compared to the huge amount of features $(p)$. This "$n<<p$" property has hampered application of deep learning techniques for disease outcome classification. Sparse learning by incorporating ex…

PCGs encompass a broader range of neural networks.

We show that there is a simple (approximately radial) function on $\reals^d$, expressible by a small 3-layer feedforward neural networks, which cannot be approximated by any 2-layer network, to more than a certain constant accuracy, unless its width is exponential in the dimension. The result holds for virtually all kn…

We propose a novel learning method for multilayered neural networks which uses feedforward supervisory signal and associates classification of a new input with that of pre-trained input. The proposed method effectively uses rich input information in the earlier layer for robust leaning and revising internal representat…

Deep neural network improves Heston model calibration accuracy and speed.

Many real world stochastic control problems suffer from the "curse of dimensionality". To overcome this difficulty, we develop a deep learning approach that directly solves high-dimensional stochastic control problems based on Monte-Carlo sampling. We approximate the time-dependent controls as feedforward neural networ…

Deep learning improves portfolio optimization efficiency.

The approximation power of general feedforward neural networks with piecewise linear activation functions is investigated. First, lower bounds on the size of a network are established in terms of the approximation error and network depth and width. These bounds improve upon state-of-the-art bounds for certain classes o…

Optimizes wide low-rank neural networks for reduced parameters and cost.

It is often hypothesized that a crucial role for recurrent connections in the brain is to constrain the set of possible response patterns, thereby shaping the neural code. This implies the existence of neural codes that cannot arise solely from feedforward processing. We set out to find such codes in the context of one…

Paper analyzes deep neural networks with dependent data, establishing convergence rates and error bounds.

While the backpropagation of error algorithm enables deep neural network training, it implies (i) bidirectional synaptic weight transport and (ii) update locking until the forward and backward passes are completed. Not only do these constraints preclude biological plausibility, but they also hinder the development of l…