Input Sparsity and Hardness for Robust Subspace Approximation

In the subspace approximation problem, we seek a k-dimensional subspace F of R^d that minimizes the sum of p-th powers of Euclidean distances to a given set of n points a_1, ..., a_n in R^d, for p >= 1. More generally than minimizing sum_i dist(a_i,F)^p,we may wish to minimize sum_i M(dist(a_i,F)) for some loss function M(), for example, M-Estimators, which include the Huber and Tukey loss functions. Such subspaces provide alternatives to the singular value decomposition (SVD), which is the p=2 case, finding such an F that minimizes the sum of squares of distances. For p in [1,2), and for typical M-Estimators, the minimizing $F$ gives a solution that is more robust to outliers than that provided by the SVD. We give several algorithmic and hardness results for these robust subspace approximation problems. We think of the n points as forming an n x d matrix A, and letting nnz(A) denote the number of non-zero entries of A. Our results hold for p in [1,2). We use poly(n) to denote n^{O(1)} as n -> infty. We obtain: (1) For minimizing sum_i dist(a_i,F)^p, we give an algorithm running in O(nnz(A) + (n+d)poly(k/eps) + exp(poly(k/eps))), (2) we show that the problem of minimizing sum_i dist(a_i, F)^p is NP-hard, even to output a (1+1/poly(d))-approximation, answering a question of Kannan and Vempala, and complementing prior results which held for p >2, (3) For loss functions for a wide class of M-Estimators, we give a problem-size reduction: for a parameter K=(log n)^{O(log k)}, our reduction takes O(nnz(A) log n + (n+d) poly(K/eps)) time to reduce the problem to a constrained version involving matrices whose dimensions are poly(K eps^{-1} log n). We also give bicriteria solutions, (4) Our techniques lead to the first O(nnz(A) + poly(d/eps)) time algorithms for (1+eps)-approximate regression for a wide class of convex M-Estimators.Comment: paper appeared in FOCS, 201

The importance of radiation as a means of controlling surface and internal temperatures of a semi-infinite solid

This problem is concerned with the temperature history of a semi-infinite solid bounded by one flat surface. Heat was applied by convection from a hot fluid to the flat surface of the solid which was originally at a uniform temperature. Heat was radiated from the flat surface to black space and conducted into the solid according to Fourier\u27s law. The temperature at the surface, and at several points beneath the surface, was determined as a function of time. A method for solution of such a problem is demonstrated here by the use of numerical analysis and the Royal McBee LGP 30 Digital Computer. The solution is applicable to all similar problems. The result demonstrates that radiation from the flat surface of the solid is very important because it results in depressing the surface temperature far below that of the hot fluid in contact with its one surface --Abstract, page iii

ADA and Title VI for the Elected Official

This session provides practical steps to assist elected officials in removing barriers to accessibility and evaluating their programs for potential discrimination. Once plans and policies are in place, it can be challenging to get these programs moving, and even more challenging to sustain them over the long term. INDOT, FHWA, and the Indiana Title VI and ADA Coordinators’ Association will connect attendees to available resources for administering these programs and maintaining compliance with the Title VI and ADA requirements

The NItty Gritty: INDOT and Title VI

This presentation provides Title VI requirements for Indiana local public agencies. Ken Woodruff opens the session with the history of Title VI and why we care. Erin Hall discusses the specifics of EJ and LEP within Title VI and what they look for. Cathy Gross provides information on networks and resources for LPAs

Sharper Bounds for Regularized Data Fitting

We study matrix sketching methods for regularized variants of linear regression, low rank approximation, and canonical correlation analysis. Our main focus is on sketching techniques which preserve the objective function value for regularized problems, which is an area that has remained largely unexplored. We study regularization both in a fairly broad setting, and in the specific context of the popular and widely used technique of ridge regularization; for the latter, as applied to each of these problems, we show algorithmic resource bounds in which the statistical dimension appears in places where in previous bounds the rank would appear. The statistical dimension is always smaller than the rank, and decreases as the amount of regularization increases. In particular we show this for the ridge low-rank approximation problem as well as regularized low-rank approximation problems in a much more general setting, where the regularizing function satisfies some very general conditions (chiefly, invariance under orthogonal transformations)

Fine Particulate Matter (PM(2.5)) Air Pollution and Selected Causes of Postneonatal Infant Mortality in California

Studies suggest that airborne particulate matter (PM) may be associated with postneonatal infant mortality, particularly with respiratory causes and sudden infant death syndrome (SIDS). To further explore this issue, we examined the relationship between long-term exposure to fine PM air pollution and postneonatal infant mortality in California. We linked monitoring data for PM ≤2.5 μm in aerodynamic diameter (PM(2.5)) to infants born in California in 1999 and 2000 using maternal addresses for mothers who lived within 5 miles of a PM(2.5) monitor. We matched each postneonatal infant death to four infants surviving to 1 year of age, by birth weight category and date of birth (within 2 weeks). For each matched set, we calculated exposure as the average PM(2.5) concentration over the period of life for the infant who died. We used conditional logistic regression to estimate the odds of postneonatal all-cause, respiratory-related, SIDS, and external-cause (a control category) mortality by exposure to PM(2.5), controlling for the matched sets and maternal demographic factors. We matched 788 postneonatal infant deaths to 3,089 infant survivors, with 51 and 120 postneonatal deaths due to respiratory causes and SIDS, respectively. We found an adjusted odds ratio for a 10−μg/m(3) increase in PM(2.5) of 1.07 [95% confidence interval (CI), 0.93–1.24] for overall postneonatal mortality, 2.13 (95% CI, 1.12–4.05) for respiratory-related postneonatal mortality, 0.82 (95% CI, 0.55–1.23) for SIDS, and 0.83 (95% CI, 0.50–1.39) for external causes. The California findings add further evidence of a PM air pollution effect on respiratory-related postneonatal infant mortality

Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra

We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression that are comparable to their quantum analogues. De-quantizing such algorithms has received a flurry of attention in recent years; we obtain sharper bounds for these problems. More significantly, we achieve these improvements by arguing that the previous quantum-inspired algorithms for these problems are doing leverage or ridge-leverage score sampling in disguise; these are powerful and standard techniques in randomized numerical linear algebra. With this recognition, we are able to employ the large body of work in numerical linear algebra to obtain algorithms for these problems that are simpler or faster (or both) than existing approaches.Comment: Adding new numerical experiment