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

Is Schumpeterian "Creative Destruction" a Plausible Source of Endogenous Real Business Cycle Shocks?

This paper looks at the linkages between growth and business cycles by bringing together two strands of literature. We incorporate a quality ladders engine of growth into an otherwise standard real business cycle model. Our fundamental question is, can Schumpeter’s creative destruction process which leads to technological improvement over time also generate realistic business cycles? We use a standard real business cycle approach to solve for rules of motion in our state variables and proceed to generate artificial time series. We compare the statistical properties of these series with their historical counterparts to determine if the model mimics the real world closely. One advantage our approach has over the standard approach is that the trend component is included in our artificial series just as it is in the data. Hence, we are not tied to any particular filtering method when we compare simulations with the real world data. Quantitative analysis reveals the model is at least as capable of accounting for key features of fluctuations at various frequencies as a model with exogenous technology shocks. Moreover, the model can do so without relying as heavily on a highly persistent generating process for such exogenous shocks as standard models must.Schumpeter, growth, cycles, real business cycles