Feature hashing and other random projection schemes are commonly used to reduce the dimensionality of feature vectors. The goal is to efficiently project a high-dimensional feature vector living in R^n into a much lower-dimensional space R^m, while approximately preserving Euclidean norm. These schemes can be constructed using sparse random projections, for example using a sparse Johnson-Lindenstrauss (JL) transform. A line of work introduced by Weinberger et. al (ICML '09) analyzes the accuracy of sparse JL with sparsity 1 on feature vectors with small linfinity-to-l2 norm ratio. Recently, Freksen, Kamma, and Larsen (NeurIPS '18) closed this line of work by proving a tight tradeoff between linfinity-to-l2 norm ratio and accuracy for sparse JL with sparsity 1. In this paper, we demonstrate the benefits of using sparsity s greater than 1 in sparse JL on feature vectors. Our main result is a tight tradeoff between linfinity-to-l2 norm ratio and accuracy for a general sparsity s, that significantly generalizes the result of Freksen et. al. Our result theoretically demonstrates that sparse JL with s > 1 can have significantly better norm-preservation properties on feature vectors than sparse JL with s = 1; we also empirically demonstrate this finding.

We study the multi-channel sparse blind deconvolution (MCS-BD) problem, whose task is to simultaneously recover a kernel $\mathbf a$ and multiple sparse inputs $\{\mathbf x_i\}_{i=1}^p$ from their circulant convolution $\mathbf y_i = \mb a \circledast \mb x_i $ ($i=1,\cdots,p$). We formulate the task as a nonconvex optimization problem over the sphere. Under mild statistical assumptions of the data, we prove that the vanilla Riemannian gradient descent (RGD) method, with random initializations, provably recovers both the kernel $\mathbf a$ and the signals $\{\mathbf x_i\}_{i=1}^p$ up to a signed shift ambiguity. In comparison with state-of-the-art results, our work shows significant improvements in terms of sample complexity and computational efficiency. Our theoretical results are corroborated by numerical experiments, which demonstrate superior performance of the proposed approach over the previous methods on both synthetic and real datasets.

Learning set functions is a key challenge arising in many domains, ranging from sketching graphs to black-box optimization with discrete parameters. In this paper we consider the problem of efficiently learning set functions that are defined over a ground set of size $n$ and that are sparse (say $k$-sparse) in the Fourier domain. This is a wide class, that includes graph and hypergraph cut functions, decision trees and more. Our central contribution is the first algorithm that allows learning functions whose Fourier support only contains low degree (say degree $d=o(n)$) polynomials using $O(k d \log n)$ sample complexity and runtime $O( kn \log^2 k \log n \log d)$. This implies that sparse graphs with $k$ edges can, for the first time, be learned from $O(k \log n)$ observations of cut values and in linear time in the number of vertices. Our algorithm can also efficiently learn (sums of) decision trees of small depth. The algorithm exploits techniques from the sparse Fourier transform literature and is easily implementable. Lastly, we also develop an efficient robust version of our algorithm and prove $\ell_2/\ell_2$ approximation guarantees without any statistical assumptions on the noise.

Compressive learning is an effective method to deal with very high dimensional datasets by applying learning algorithms in a randomly projected lower dimensional space. In this paper, we consider the learning problem where the projected data is further compressed by scalar quantization, which is called quantized compressive learning. Generalization error bounds are derived for three models: nearest neighbor (NN) classifier, linear classifier and least squares regression. Besides studying finite sample setting, our asymptotic analysis shows that the inner product estimators have deep connection with NN and linear classification problem through the variance of their debiased counterparts. By analyzing the extra error term brought by quantization, our results provide useful implications to the choice of quantizers in applications involving different learning tasks. Empirical study is also conducted to validate our theoretical findings.

We investigate a sequential optimization procedure to minimize the empirical risk functional $f_{\hat\theta}(x) = \frac{1}{2}\|G_{\hat\theta}(x) - y\|^2$ for certain families of deep networks $G_{\theta}(x)$. The approach is to optimize a sequence of objective functions that use network parameters obtained during different stages of the training process. When initialized with random parameters $\theta_0$, we show that the objective $f_{\theta_0}(x)$ is ``nice'' and easy to optimize with gradient descent. As learning is carried out, we obtain a sequence of generative networks $x \mapsto G_{\theta_t}(x)$ and associated risk functions $f_{\theta_t}(x)$, where $t$ indicates a stage of stochastic gradient descent during training. Since the parameters of the network do not change by very much in each step, the surface evolves slowly and can be incrementally optimized. The algorithm is formalized and analyzed for a family of expansive networks. We call the procedure {\it surfing} since it rides along the peak of the evolving (negative) empirical risk function, starting from a smooth surface at the beginning of learning and ending with a wavy nonconvex surface after learning is complete. Experiments show how surfing can be used to find the global optimum and for compressed sensing even when direct gradient descent on the final learned network fails.

Interest point detection and local feature description are fundamental steps in many computer vision applications. Classical approaches are based on a detect-then-describe paradigm where separate handcrafted methods are used to first identify repeatable keypoints and then represent them with a local descriptor. Neural networks trained with metric learning losses have recently caught up with these techniques, focusing on learning repeatable saliency maps for keypoint detection or learning descriptors at the detected keypoint locations. In this work, we argue that repeatable regions are not necessarily discriminative and can therefore lead to select suboptimal keypoints. Furthermore, we claim that descriptors should be learned only in regions for which matching can be performed with high confidence. We thus propose to jointly learn keypoint detection and description together with a predictor of the local descriptor discriminativeness. This allows to avoid ambiguous areas, thus leading to reliable keypoint detection and description. Our detection-and-description approach simultaneously outputs sparse, repeatable and reliable keypoints that outperforms state-of-the-art detectors and descriptors on the HPatches dataset and on the recent Aachen Day-Night localization benchmark.

Video interpolation is an important problem in computer vision, which helps overcome the temporal limitation of camera sensors. Existing video interpolation methods usually assume uniform motion between consecutive frames and use linear models for interpolation, which cannot well approximate the complex motion in the real world. To address these issues, we propose a quadratic video interpolation method which exploits the acceleration information in videos. This method allows prediction with curvilinear trajectory and variable velocity, and generates more accurate interpolation results. For high-quality frame synthesis, we develop a flow reversal layer to estimate flow fields starting from the unknown target frame to the source frame. In addition, we present techniques for flow refinement. Extensive experiments demonstrate that our approach performs favorably against the existing linear models on a wide variety of video datasets.

Deep neural networks have been very successful in compressive-sensing and image restoration applications, as a means to estimate images from partial, blurry, or otherwise degraded measurements. These networks are trained on a large number of corresponding pairs of measurements and ground-truth images, and thus implicitly learn to exploit domain-specific image statistics. But unlike measurement data, it is often expensive or impractical to collect a large training set of ground-truth images in many application settings. In this paper, we introduce an unsupervised framework for training image estimation networks, from a training set that contains only measurements---with two varied measurements per image---but no ground-truth for the full images desired as output. We demonstrate that our framework can be applied for both regular and blind image estimation tasks, where in the latter case parameters of the measurement model (e.g., the blur kernel) are unknown: during inference, and potentially, also during training. We evaluate our framework for training networks for compressive-sensing and blind deconvolution, considering both non-blind and blind training for the latter. Our framework yields models that are nearly as accurate as those from fully supervised training, despite not having access to any ground-truth images.

In this paper, we introduce STREETS, a novel traffic flow dataset from publicly available web cameras in the suburbs of Chicago, IL. We seek to address the limitations of existing datasets in this area. Many such datasets lack a coherent traffic network graph to describe the relationship between sensors. The datasets that do provide a graph depict traffic flow in urban population centers or highway systems and use costly sensors like induction loops. These contexts differ from that of a suburban traffic body. Our dataset provides over 4 million still images across 2.5 months and one hundred web cameras in suburban Lake County, IL. We divide the cameras into two distinct communities described by directed graphs and count vehicles to track traffic statistics. Our goal is to give researchers a benchmark dataset for exploring the capabilities of inexpensive and non-invasive sensors like web cameras to understand complex traffic bodies in communities of any size. We present benchmarking tasks and baseline results for one such task to guide how future work may use our dataset.

When we take photos through glass windows or doors, the transmitted background scene is often blended with undesirable reflection. Separating two layers apart to enhance the image quality is of vital importance for both human and machine perception. In this paper, we propose to exploit physical constraints from a pair of unpolarized and polarized images to separate reflection and transmission layers. Due to the simplified capturing setup, the system becomes more underdetermined compared with existing polarization based solutions that take three or more images as input. We propose to solve semireflector orientation estimation first to make the physical image formation well-posed and then learn to reliably separate two layers using a refinement network with gradient loss. Quantitative and qualitative experimental results show our approach performs favorably over existing polarization and single image based solutions.