GraphConnect: A Regularization Framework for Neural Networks

Deep neural networks have proved very successful in domains where large training sets are available, but when the number of training samples is small, their performance suffers from overfitting. Prior methods of reducing overfitting such as weight decay, Dropout and DropConnect are data-independent. This paper proposes a new method, GraphConnect, that is data-dependent, and is motivated by the observation that data of interest lie close to a manifold. The new method encourages the relationships between the learned decisions to resemble a graph representing the manifold structure. Essentially GraphConnect is designed to learn attributes that are present in data samples in contrast to weight decay, Dropout and DropConnect which are simply designed to make it more difficult to fit to random error or noise. Empirical Rademacher complexity is used to connect the generalization error of the neural network to spectral properties of the graph learned from the input data. This framework is used to show that GraphConnect is superior to weight decay. Experimental results on several benchmark datasets validate the theoretical analysis, and show that when the number of training samples is small, GraphConnect is able to significantly improve performance over weight decay.

Hand-held Video Deblurring via Efficient Fourier Aggregation

Videos captured with hand-held cameras often suffer from a significant amount of blur, mainly caused by the inevitable natural tremor of the photographer's hand. In this work, we present an algorithm that removes blur due to camera shake by combining information in the Fourier domain from nearby frames in a video. The dynamic nature of typical videos with the presence of multiple moving objects and occlusions makes this problem of camera shake removal extremely challenging, in particular when low complexity is needed. Given an input video frame, we first create a consistent registered version of temporally adjacent frames. Then, the set of consistently registered frames is block-wise fused in the Fourier domain with weights depending on the Fourier spectrum magnitude. The method is motivated from the physiological fact that camera shake blur has a random nature and therefore, nearby video frames are generally blurred differently. Experiments with numerous videos recorded in the wild, along with extensive comparisons, show that the proposed algorithm achieves state-of-the-art results while at the same time being much faster than its competitors.

Removing Camera Shake via Weighted Fourier Burst Accumulation

Numerous recent approaches attempt to remove image blur due to camera shake, either with one or multiple input images, by explicitly solving an inverse and inherently ill-posed deconvolution problem. If the photographer takes a burst of images, a modality available in virtually all modern digital cameras, we show that it is possible to combine them to get a clean sharp version. This is done without explicitly solving any blur estimation and subsequent inverse problem. The proposed algorithm is strikingly simple: it performs a weighted average in the Fourier domain, with weights depending on the Fourier spectrum magnitude. The method can be seen as a generalization of the align and average procedure, with a weighted average, motivated by hand-shake physiology and theoretically supported, taking place in the Fourier domain. The method's rationale is that camera shake has a random nature and therefore each image in the burst is generally blurred differently. Experiments with real camera data, and extensive comparisons, show that the proposed Fourier Burst Accumulation (FBA) algorithm achieves state-of-the-art results an order of magnitude faster, with simplicity for on-board implementation on camera phones. Finally, we also present experiments in real high dynamic range (HDR) scenes, showing how the method can be straightforwardly extended to HDR photography.

Geometry-aware Deep Transform

Many recent efforts have been devoted to designing sophisticated deep learning structures, obtaining revolutionary results on benchmark datasets. The success of these deep learning methods mostly relies on an enormous volume of labeled training samples to learn a huge number of parameters in a network; therefore, understanding the generalization ability of a learned deep network cannot be overlooked, especially when restricted to a small training set, which is the case for many applications. In this paper, we propose a novel deep learning objective formulation that unifies both the classification and metric learning criteria. We then introduce a geometry-aware deep transform to enable a non-linear discriminative and robust feature transform, which shows competitive performance on small training sets for both synthetic and real-world data. We further support the proposed framework with a formal $(K,\epsilon)$-robustness analysis.

Compressed Nonnegative Matrix Factorization is Fast and Accurate

Nonnegative matrix factorization (NMF) has an established reputation as a useful data analysis technique in numerous applications. However, its usage in practical situations is undergoing challenges in recent years. The fundamental factor to this is the increasingly growing size of the datasets available and needed in the information sciences. To address this, in this work we propose to use structured random compression, that is, random projections that exploit the data structure, for two NMF variants: classical and separable. In separable NMF (SNMF) the left factors are a subset of the columns of the input matrix. We present suitable formulations for each problem, dealing with different representative algorithms within each one. We show that the resulting compressed techniques are faster than their uncompressed variants, vastly reduce memory demands, and do not encompass any significant deterioration in performance. The proposed structured random projections for SNMF allow to deal with arbitrarily shaped large matrices, beyond the standard limit of tall-and-skinny matrices, granting access to very efficient computations in this general setting. We accompany the algorithmic presentation with theoretical foundations and numerous and diverse examples, showing the suitability of the proposed approaches.

Data Representation using the Weyl Transform

The Weyl transform is introduced as a rich framework for data representation. Transform coefficients are connected to the Walsh-Hadamard transform of multiscale autocorrelations, and different forms of dyadic periodicity in a signal are shown to appear as different features in its Weyl coefficients. The Weyl transform has a high degree of symmetry with respect to a large group of multiscale transformations, which allows compact yet discriminative representations to be obtained by pooling coefficients. The effectiveness of the Weyl transform is demonstrated through the example of textured image classification.

On the Stability of Deep Networks

In this work we study the properties of deep neural networks (DNN) with random weights. We formally prove that these networks perform a distance-preserving embedding of the data. Based on this we then draw conclusions on the size of the training data and the networks' structure. A longer version of this paper with more results and details can be found in (Giryes et al., 2015). In particular, we formally prove in the longer version that DNN with random Gaussian weights perform a distance-preserving embedding of the data, with a special treatment for in-class and out-of-class data.

Random Forests Can Hash

Hash codes are a very efficient data representation needed to be able to cope with the ever growing amounts of data. We introduce a random forest semantic hashing scheme with information-theoretic code aggregation, showing for the first time how random forest, a technique that together with deep learning have shown spectacular results in classification, can also be extended to large-scale retrieval. Traditional random forest fails to enforce the consistency of hashes generated from each tree for the same class data, i.e., to preserve the underlying similarity, and it also lacks a principled way for code aggregation across trees. We start with a simple hashing scheme, where independently trained random trees in a forest are acting as hashing functions. We the propose a subspace model as the splitting function, and show that it enforces the hash consistency in a tree for data from the same class. We also introduce an information-theoretic approach for aggregating codes of individual trees into a single hash code, producing a near-optimal unique hash for each class. Experiments on large-scale public datasets are presented, showing that the proposed approach significantly outperforms state-of-the-art hashing methods for retrieval tasks.

Graph Matching: Relax at Your Own Risk

Graph matching---aligning a pair of graphs to minimize their edge disagreements---has received wide-spread attention from both theoretical and applied communities over the past several decades, including combinatorics, computer vision, and connectomics. Its attention can be partially attributed to its computational difficulty. Although many heuristics have previously been proposed in the literature to approximately solve graph matching, very few have any theoretical support for their performance. A common technique is to relax the discrete problem to a continuous problem, therefore enabling practitioners to bring gradient-descent-type algorithms to bear. We prove that an indefinite relaxation (when solved exactly) almost always discovers the optimal permutation, while a common convex relaxation almost always fails to discover the optimal permutation. These theoretical results suggest that initializing the indefinite algorithm with the convex optimum might yield improved practical performance. Indeed, experimental results illuminate and corroborate these theoretical findings, demonstrating that excellent results are achieved in both benchmark and real data problems by amalgamating the two approaches.

A Bi-clustering Framework for Consensus Problems

We consider grouping as a general characterization for problems such as clustering, community detection in networks, and multiple parametric model estimation. We are interested in merging solutions from different grouping algorithms, distilling all their good qualities into a consensus solution. In this paper, we propose a bi-clustering framework and perspective for reaching consensus in such grouping problems. In particular, this is the first time that the task of finding/fitting multiple parametric models to a dataset is formally posed as a consensus problem. We highlight the equivalence of these tasks and establish the connection with the computational Gestalt program, that seeks to provide a psychologically-inspired detection theory for visual events. We also present a simple but powerful bi-clustering algorithm, specially tuned to the nature of the problem we address, though general enough to handle many different instances inscribed within our characterization. The presentation is accompanied with diverse and extensive experimental results in clustering, community detection, and multiple parametric model estimation in image processing applications.