We study the geometry of deep (neural) networks (DNs) with piecewise affine and convex nonlinearities. The layers of such DNs have been shown to be {\em max-affine spline operators} (MASOs) that partition their input space and apply a region-dependent affine mapping to their input to produce their output. We demonstrate that each MASO layer's input space partitioning corresponds to a {\em power diagram} (an extension of the classical Voronoi tiling) with a number of regions that grows exponentially with respect to the number of units (neurons). We further show that a composition of MASO layers (e.g., the entire DN) produces a progressively subdivided power diagram and provide its analytical form. The subdivision process constrains the affine maps on the (exponentially many) power diagram regions to greatly reduce their complexity. For classification problems, we obtain a formula for a MASO DN's decision boundary in the input space plus a measure of its curvature that depends on the DN's nonlinearities, weights, and architecture. Numerous numerical experiments support and extend our theoretical results.
We build a rigorous bridge between deep networks (DNs) and approximation theory via spline functions and operators. Our key result is that a large class of DNs can be written as a composition of max-affine spline operators (MASOs), which provide a powerful portal through which to view and analyze their inner workings. For instance, conditioned on the input signal, the output of a MASO DN can be written as a simple affine transformation of the input. This implies that a DN constructs a set of signal-dependent, class-specific templates against which the signal is compared via a simple inner product; we explore the links to the classical theory of optimal classification via matched filters and the effects of data memorization. Going further, we propose a simple penalty term that can be added to the cost function of any DN learning algorithm to force the templates to be orthogonal with each other; this leads to significantly improved classification performance and reduced overfitting with no change to the DN architecture. The spline partition of the input signal space that is implicitly induced by a MASO directly links DNs to the theory of vector quantization (VQ) and $K$-means clustering, which opens up new geometric avenue to study how DNs organize signals in a hierarchical fashion. To validate the utility of the VQ interpretation, we develop and validate a new distance metric for signals and images that quantifies the difference between their VQ encodings. (This paper is a significantly expanded version of A Spline Theory of Deep Learning from ICML 2018.)
Deep Neural Networks (DNNs) provide state-of-the-art solutions in several difficult machine perceptual tasks. However, their performance relies on the availability of a large set of labeled training data, which limits the breadth of their applicability. Hence, there is a need for new {\em semi-supervised learning} methods for DNNs that can leverage both (a small amount of) labeled and unlabeled training data. In this paper, we develop a general loss function enabling DNNs of any topology to be trained in a semi-supervised manner without extra hyper-parameters. As opposed to current semi-supervised techniques based on topology-specific or unstable approaches, ours is both robust and general. We demonstrate that our approach reaches state-of-the-art performance on the SVHN ($9.82\%$ test error, with $500$ labels and wide Resnet) and CIFAR10 (16.38% test error, with 8000 labels and sigmoid convolutional neural network) data sets.
In this work, we derive a generic overcomplete frame thresholding scheme based on risk minimization. Overcomplete frames being favored for analysis tasks such as classification, regression or anomaly detection, we provide a way to leverage those optimal representations in real-world applications through the use of thresholding. We validate the method on a large scale bird activity detection task via the scattering network architecture performed by means of continuous wavelets, known for being an adequate dictionary in audio environments.
Deep Neural Networks (DNNs) are universal function approximators providing state-of- the-art solutions on wide range of applications. Common perceptual tasks such as speech recognition, image classification, and object tracking are now commonly tackled via DNNs. Some fundamental problems remain: (1) the lack of a mathematical framework providing an explicit and interpretable input-output formula for any topology, (2) quantification of DNNs stability regarding adversarial examples (i.e. modified inputs fooling DNN predictions whilst undetectable to humans), (3) absence of generalization guarantees and controllable behaviors for ambiguous patterns, (4) leverage unlabeled data to apply DNNs to domains where expert labeling is scarce as in the medical field. Answering those points would provide theoretical perspectives for further developments based on a common ground. Furthermore, DNNs are now deployed in tremendous societal applications, pushing the need to fill this theoretical gap to ensure control, reliability, and interpretability.
FlatCam is a thin form-factor lensless camera that consists of a coded mask placed on top of a bare, conventional sensor array. Unlike a traditional, lens-based camera where an image of the scene is directly recorded on the sensor pixels, each pixel in FlatCam records a linear combination of light from multiple scene elements. A computational algorithm is then used to demultiplex the recorded measurements and reconstruct an image of the scene. FlatCam is an instance of a coded aperture imaging system; however, unlike the vast majority of related work, we place the coded mask extremely close to the image sensor that can enable a thin system. We employ a separable mask to ensure that both calibration and image reconstruction are scalable in terms of memory requirements and computational complexity. We demonstrate the potential of the FlatCam design using two prototypes: one at visible wavelengths and one at infrared wavelengths.
Sparse approximations using highly over-complete dictionaries is a state-of-the-art tool for many imaging applications including denoising, super-resolution, compressive sensing, light-field analysis, and object recognition. Unfortunately, the applicability of such methods is severely hampered by the computational burden of sparse approximation: these algorithms are linear or super-linear in both the data dimensionality and size of the dictionary. We propose a framework for learning the hierarchical structure of over-complete dictionaries that enables fast computation of sparse representations. Our method builds on tree-based strategies for nearest neighbor matching, and presents domain-specific enhancements that are highly efficient for the analysis of image patches. Contrary to most popular methods for building spatial data structures, out methods rely on shallow, balanced trees with relatively few layers. We show an extensive array of experiments on several applications such as image denoising/superresolution, compressive video/light-field sensing where we practically achieve 100-1000x speedup (with a less than 1dB loss in accuracy).
Compressed sensing enables the reconstruction of high-resolution signals from under-sampled data. While compressive methods simplify data acquisition, they require the solution of difficult recovery problems to make use of the resulting measurements. This article presents a new sensing framework that combines the advantages of both conventional and compressive sensing. Using the proposed \stone transform, measurements can be reconstructed instantly at Nyquist rates at any power-of-two resolution. The same data can then be "enhanced" to higher resolutions using compressive methods that leverage sparsity to "beat" the Nyquist limit. The availability of a fast direct reconstruction enables compressive measurements to be processed on small embedded devices. We demonstrate this by constructing a real-time compressive video camera.