Matrix completion, where we wish to recover a low rank matrix by observing a few entries from it, is a widely studied problem in both theory and practice with wide applications. Most of the provable algorithms so far on this problem have been restricted to the offline setting where they provide an estimate of the unknown matrix using all observations simultaneously. However, in many applications, the online version, where we observe one entry at a time and dynamically update our estimate, is more appealing. While existing algorithms are efficient for the offline setting, they could be highly inefficient for the online setting. In this paper, we propose the first provable, efficient online algorithm for matrix completion. Our algorithm starts from an initial estimate of the matrix and then performs non-convex stochastic gradient descent (SGD). After every observation, it performs a fast update involving only one row of two tall matrices, giving near linear total runtime. Our algorithm can be naturally used in the offline setting as well, where it gives competitive sample complexity and runtime to state of the art algorithms. Our proofs introduce a general framework to show that SGD updates tend to stay away from saddle surfaces and could be of broader interests for other non-convex problems to prove tight rates.

We introduce a new representation and feature extraction method for biological sequences. Named bio-vectors (BioVec) to refer to biological sequences in general with protein-vectors (ProtVec) for proteins (amino-acid sequences) and gene-vectors (GeneVec) for gene sequences, this representation can be widely used in applications of deep learning in proteomics and genomics. In the present paper, we focus on protein-vectors that can be utilized in a wide array of bioinformatics investigations such as family classification, protein visualization, structure prediction, disordered protein identification, and protein-protein interaction prediction. In this method, we adopt artificial neural network approaches and represent a protein sequence with a single dense n-dimensional vector. To evaluate this method, we apply it in classification of 324,018 protein sequences obtained from Swiss-Prot belonging to 7,027 protein families, where an average family classification accuracy of 93%+-0.06% is obtained, outperforming existing family classification methods. In addition, we use ProtVec representation to predict disordered proteins from structured proteins. Two databases of disordered sequences are used: the DisProt database as well as a database featuring the disordered regions of nucleoporins rich with phenylalanine-glycine repeats (FG-Nups). Using support vector machine classifiers, FG-Nup sequences are distinguished from structured protein sequences found in Protein Data Bank (PDB) with a 99.8% accuracy, and unstructured DisProt sequences are differentiated from structured DisProt sequences with 100.0% accuracy. These results indicate that by only providing sequence data for various proteins into this model, accurate information about protein structure can be determined.

Identification of regions of interest (ROI) associated with certain disease has a great impact on public health. Imposing sparsity of pixel values and extracting active regions simultaneously greatly complicate the image analysis. We address these challenges by introducing a novel region-selection penalty in the framework of image-on-scalar regression. Our penalty combines the Smoothly Clipped Absolute Deviation (SCAD) regularization, enforcing sparsity, and the SCAD of total variation (TV) regularization, enforcing spatial contiguity, into one group, which segments contiguous spatial regions against zero-valued background. Efficient algorithm is based on the alternative direction method of multipliers (ADMM) which decomposes the non-convex problem into two iterative optimization problems with explicit solutions. Another virtue of the proposed method is that a divide and conquer learning algorithm is developed, thereby allowing scaling to large images. Several examples are presented and the experimental results are compared with other state-of-the-art approaches.

Performance of nuclear threat detection systems based on gamma-ray spectrometry often strongly depends on the ability to identify the part of measured signal that can be attributed to background radiation. We have successfully applied a method based on Principal Component Analysis (PCA) to obtain a compact null-space model of background spectra using PCA projection residuals to derive a source detection score. We have shown the method's utility in a threat detection system using mobile spectrometers in urban scenes (Tandon et al 2012). While it is commonly assumed that measured photon counts follow a Poisson process, standard PCA makes a Gaussian assumption about the data distribution, which may be a poor approximation when photon counts are low. This paper studies whether and in what conditions PCA with a Poisson-based loss function (Poisson PCA) can outperform standard Gaussian PCA in modeling background radiation to enable more sensitive and specific nuclear threat detection.

We study the generalization properties of stochastic gradient methods for learning with convex loss functions and linearly parameterized functions. We show that, in the absence of penalizations or constraints, the stability and approximation properties of the algorithm can be controlled by tuning either the step-size or the number of passes over the data. In this view, these parameters can be seen to control a form of implicit regularization. Numerical results complement the theoretical findings.

We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms for batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.

We show that there are no spurious local minima in the non-convex factorized parametrization of low-rank matrix recovery from incoherent linear measurements. With noisy measurements we show all local minima are very close to a global optimum. Together with a curvature bound at saddle points, this yields a polynomial time global convergence guarantee for stochastic gradient descent {\em from random initialization}.

Topic models have emerged as fundamental tools in unsupervised machine learning. Most modern topic modeling algorithms take a probabilistic view and derive inference algorithms based on Latent Dirichlet Allocation (LDA) or its variants. In contrast, we study topic modeling as a combinatorial optimization problem, and propose a new objective function derived from LDA by passing to the small-variance limit. We minimize the derived objective by using ideas from combinatorial optimization, which results in a new, fast, and high-quality topic modeling algorithm. In particular, we show that our results are competitive with popular LDA-based topic modeling approaches, and also discuss the (dis)similarities between our approach and its probabilistic counterparts.

Classifier evasion consists in finding for a given instance $x$ the nearest instance $x'$ such that the classifier predictions of $x$ and $x'$ are different. We present two novel algorithms for systematically computing evasions for tree ensembles such as boosted trees and random forests. Our first algorithm uses a Mixed Integer Linear Program solver and finds the optimal evading instance under an expressive set of constraints. Our second algorithm trades off optimality for speed by using symbolic prediction, a novel algorithm for fast finite differences on tree ensembles. On a digit recognition task, we demonstrate that both gradient boosted trees and random forests are extremely susceptible to evasions. Finally, we harden a boosted tree model without loss of predictive accuracy by augmenting the training set of each boosting round with evading instances, a technique we call adversarial boosting.

We introduce a new representation learning approach for domain adaptation, in which data at training and test time come from similar but different distributions. Our approach is directly inspired by the theory on domain adaptation suggesting that, for effective domain transfer to be achieved, predictions must be made based on features that cannot discriminate between the training (source) and test (target) domains. The approach implements this idea in the context of neural network architectures that are trained on labeled data from the source domain and unlabeled data from the target domain (no labeled target-domain data is necessary). As the training progresses, the approach promotes the emergence of features that are (i) discriminative for the main learning task on the source domain and (ii) indiscriminate with respect to the shift between the domains. We show that this adaptation behaviour can be achieved in almost any feed-forward model by augmenting it with few standard layers and a new gradient reversal layer. The resulting augmented architecture can be trained using standard backpropagation and stochastic gradient descent, and can thus be implemented with little effort using any of the deep learning packages. We demonstrate the success of our approach for two distinct classification problems (document sentiment analysis and image classification), where state-of-the-art domain adaptation performance on standard benchmarks is achieved.