This paper investigates the phase retrieval problem, which aims to recover a signal from the magnitudes of its linear measurements. We develop statistically and computationally efficient algorithms for the situation when the measurements are corrupted by sparse outliers that can take arbitrary values. We propose a novel approach to robustify the gradient descent algorithm by using the sample median as a guide for pruning spurious samples in initialization and local search. Adopting the Poisson loss and the reshaped quadratic loss respectively, we obtain two algorithms termed median-TWF and median-RWF, both of which provably recover the signal from a near-optimal number of measurements when the measurement vectors are composed of i.i.d. Gaussian entries, up to a logarithmic factor, even when a constant fraction of the measurements are adversarially corrupted. We further show that both algorithms are stable in the presence of additional dense bounded noise. Our analysis is accomplished by developing non-trivial concentration results of median-related quantities, which may be of independent interest. We provide numerical experiments to demonstrate the effectiveness of our approach.

Recent several years have witnessed the surge of asynchronous (async-) parallel computing methods due to the extremely big data involved in many modern applications and also the advancement of multi-core machines and computer clusters. In optimization, most works about async-parallel methods are on unconstrained problems or those with block separable constraints. In this paper, we propose an async-parallel method based on block coordinate update (BCU) for solving convex problems with nonseparable linear constraint. Running on a single node, the method becomes a novel randomized primal-dual BCU with adaptive stepsize for multi-block affinely constrained problems. For these problems, Gauss-Seidel cyclic primal-dual BCU needs strong convexity to have convergence. On the contrary, merely assuming convexity, we show that the objective value sequence generated by the proposed algorithm converges in probability to the optimal value and also the constraint residual to zero. In addition, we establish an ergodic $O(1/k)$ convergence result, where $k$ is the number of iterations. Numerical experiments are performed to demonstrate the efficiency of the proposed method and significantly better speed-up performance than its sync-parallel counterpart.

We consider the problem of efficient randomized dimensionality reduction with norm-preservation guarantees. Specifically we prove data-dependent Johnson-Lindenstrauss-type geometry preservation guarantees for Ho's random subspace method: When data satisfy a mild regularity condition -- the extent of which can be estimated by sampling from the data -- then random subspace approximately preserves the Euclidean geometry of the data with high probability. Our guarantees are of the same order as those for random projection, namely the required dimension for projection is logarithmic in the number of data points, but have a larger constant term in the bound which depends upon this regularity. A challenging situation is when the original data have a sparse representation, since this implies a very large projection dimension is required: We show how this situation can be improved for sparse binary data by applying an efficient `densifying' preprocessing, which neither changes the Euclidean geometry of the data nor requires an explicit matrix-matrix multiplication. We corroborate our theoretical findings with experiments on both dense and sparse high-dimensional datasets from several application domains.

Principal Component Analysis (PCA) is a very successful dimensionality reduction technique, widely used in predictive modeling. A key factor in its widespread use in this domain is the fact that the projection of a dataset onto its first $K$ principal components minimizes the sum of squared errors between the original data and the projected data over all possible rank $K$ projections. Thus, PCA provides optimal low-rank representations of data for least-squares linear regression under standard modeling assumptions. On the other hand, when the loss function for a prediction problem is not the least-squares error, PCA is typically a heuristic choice of dimensionality reduction -- in particular for classification problems under the zero-one loss. In this paper we target classification problems by proposing a straightforward alternative to PCA that aims to minimize the difference in margin distribution between the original and the projected data. Extensive experiments show that our simple approach typically outperforms PCA on any particular dataset, in terms of classification error, though this difference is not always statistically significant, and despite being a filter method is frequently competitive with Partial Least Squares (PLS) and Lasso on a wide range of datasets.

In a series of recent works, we have generalised the consistency results in the stochastic block model literature to the case of uniform and non-uniform hypergraphs. The present paper continues the same line of study, where we focus on partitioning weighted uniform hypergraphs---a problem often encountered in computer vision. This work is motivated by two issues that arise when a hypergraph partitioning approach is used to tackle computer vision problems: (i) The uniform hypergraphs constructed for higher-order learning contain all edges, but most have negligible weights. Thus, the adjacency tensor is nearly sparse, and yet, not binary. (ii) A more serious concern is that standard partitioning algorithms need to compute all edge weights, which is computationally expensive for hypergraphs. This is usually resolved in practice by merging the clustering algorithm with a tensor sampling strategy---an approach that is yet to be analysed rigorously. We build on our earlier work on partitioning dense unweighted uniform hypergraphs (Ghoshdastidar and Dukkipati, ICML, 2015), and address the aforementioned issues by proposing provable and efficient partitioning algorithms. Our analysis justifies the empirical success of practical sampling techniques. We also complement our theoretical findings by elaborate empirical comparison of various hypergraph partitioning schemes.

We develop a new method called Discriminated Hub Graphical Lasso (DHGL) based on Hub Graphical Lasso (HGL) by providing prior information of hubs. We apply this new method in two situations: with known hubs and without known hubs. Then we compare DHGL with HGL using several measures of performance. When some hubs are known, we can always estimate the precision matrix better via DHGL than HGL. When no hubs are known, we use Graphical Lasso (GL) to provide information of hubs and find that the performance of DHGL will always be better than HGL if correct prior information is given and will seldom degenerate when the prior information is wrong.

Just look at it: it not only gives you a summary of all the specified validation measures across different clustering algorithms and number of inspected clusters, but also it lists those algorithms and number of clusters pairs that performed best in regard to a given validation metric.

Propensity score analysis is used when experimentation is not feasible or as a recourse when experiments go awry ("broken" experiments). Its basic concepts were hammered out over the span of several decades by Jerzy Neyman, William Cochrane, Donald Rubin and several other eminent statisticians, and the thinking of distinguished economist James Heckman has also influenced its development. Propensity score analysis in several variations has seen extensive use in medical research, economics, education, assessment of government programs and, more recently, in marketing research and predictive analytics. First, why do we use experiments? We may wish to test the efficacy of some treatment or intervention such as medication, therapy and counseling or, in the case of marketing, liking for a new product.

And why is it useful for BigData? 29 What is "continuous features binning"? What is a Standard Scaling? 38 Why are statistical distributions important? What is a Bias - Variance tradeoff? What is a training set, a validation set, a test set and a gold set in supervised and unsupervised learning? What is a cross-validation and what is an overfitting?

Guest blog by Kevin Gray.. Kevin is president of Cannon Gray, a marketing science and analytics consultancy. Regression is arguably the workhorse of statistics. Despite its popularity, however, it may also be the most misunderstood. The answer might surprise you: There is no such thing as Regression. The Dependent Variable is something you want to predict or explain.