Relative Errors for Deterministic Low-Rank Matrix Approximations

We consider processing an n x d matrix A in a stream with row-wise updates according to a recent algorithm called Frequent Directions (Liberty, KDD 2013). This algorithm maintains an l x d matrix Q deterministically, processing each row in O(d l^2) time; the processing time can be decreased to O(d l) with a slight modification in the algorithm and a constant increase in space. We show that if one sets l = k+ k/eps and returns Q_k, a k x d matrix that is the best rank k approximation to Q, then we achieve the following properties: ||A - A_k||_F^2 <= ||A||_F^2 - ||Q_k||_F^2 <= (1+eps) ||A - A_k||_F^2 and where pi_{Q_k}(A) is the projection of A onto the rowspace of Q_k then ||A - pi_{Q_k}(A)||_F^2 <= (1+eps) ||A - A_k||_F^2. We also show that Frequent Directions cannot be adapted to a sparse version in an obvious way that retains the l original rows of the matrix, as opposed to a linear combination or sketch of the rows.Comment: 16 pages, 0 figure

Development of a Case-Mix Funding System for Adults with Combined Vision and Hearing Loss

Background: Adults with vision and hearing loss, or dual sensory loss (DSL), present with a wide range of needs and abilities. This creates many challenges when attempting to set the most appropriate and equitable funding levels. Case-mix (CM) funding models represent one method for understanding client characteristics that correlate with resource intensity. Methods: A CM model was developed based on a derivation sample (n = 182) and tested with a replication sample (n = 135) of adults aged 18+ with known DSL who were living in the community. All items within the CM model came from a standardized, multidimensional assessment, the interRAI Community Health Assessment and the Deafblind Supplement. The main outcome was a summary of formal and informal service costs which included intervenor and interpreter support, in-home nursing, personal support and rehabilitation services. Informal costs were estimated based on a wage rate of half that for a professional service provider ($10/hour). Decision-tree analysis was used to create groups with homogeneous resource utilization. Results: The resulting CM model had 9 terminal nodes. The CM index (CMI) showed a 35-fold range for total costs. In both the derivation and replication sample, 4 groups (out of a total of 18 or 22.2%) had a coefficient of variation value that exceeded the overall level of variation. Explained variance in the derivation sample was 67.7% for total costs versus 28.2% in the replication sample. A strong correlation was observed between the CMI values in the two samples (r = 0.82; p = 0.006). Conclusions: The derived CM funding model for adults with DSL differentiates resource intensity across 9 main groups and in both datasets there is evidence that these CM groups appropriately identify clients based on need for formal and informal support

Coulomb Confinement from the Yang-Mills Vacuum State in 2+1 Dimensions

The Coulomb-gauge ghost propagator, and the color-Coulomb potential, are computed in an ensemble of configurations derived from our recently proposed Yang-Mills vacuum wavefunctional in 2+1 dimensions. The results are compared to the corresponding values obtained by standard Monte Carlo simulations in three Euclidean dimensions. The agreement is quite striking for the Coulomb-gauge ghost propagator. The color-Coulomb potential rises linearly at large distances, but its determination suffers from rather large statistical fluctuations, due to configurations with very low values of $\mu_0$, the lowest eigenvalue of the Coulomb-gauge Faddeev-Popov operator. However, if one imposes cuts on the data, effectively leaving out configurations with very low $\mu_0$, the agreement of the potential in both sets of configurations is again satisfactory, although the errorbars grow systematically as the cutoff is eliminated.Comment: 8 pages, 5 figures (10 EPS files), RevTeX4.1. V2: original figs. 4 and 5 compressed into a new fig. 5; a new fig. 4; sec. IV.B slightly modified to reflect the changes. Version to appear in Phys. Rev. D. V3: a reference corrected