We consider the problem of localizing a set of microphones together with a set of external acoustic events (e.g., hand claps), emitted at unknown times and unknown locations. We propose a solution that approximates this problem under a far field approximation defined in the calculus of affine geometry, and that relies on singular value decomposition (SVD) to recover the affine structure of the ...

NOTE: I have not thoroughly proofread these notes!!! We recommend the reader review Weeks 9-11 of Linear Algebra: Foundations to Frontiers-Notes to LAFF With.

In this report, we investigate the truncated multilinear singular value decomposition (MLSVD), proposed in De Lathauwer et al. (2000). Truncating the MLSVD results in an approximation, with a prescribed multilinear rank, to a tensor. We present a new error expression for an approximate Tucker decomposition with orthogonal factor matrices. From this expression, new insights are obtained which le...

then σ is a singular value of A and u and v are corresponding left and right singular vectors, respectively. (For generality it is assumed that the matrices here are complex, although given these results, the analogs for real matrices are obvious.) If, for a given positive singular value, there are exactly t linearly independent corresponding right singular vectors and t linearly independent co...

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The singular value decomposition, or SVD , has been studied in the past as a tool for detecting and understanding patterns in a collection of documents. We show how the matrices produced by the SVD calculation can be interpreted, allowing us to spot patterns of characters that indicate particular topics in a corpus. A test collection, consisting of two days of AP newswire tra c, is used as a ru...

We present a variation of Paige's algorithm for computing the generalized singular value decomposition (GSVD) of two matrices A and B. There are two innovations. The rst is a new preprocessing step which reduces A and B to upper triangular forms satisfying certain rank conditions. The second is a new 2 by 2 triangular GSVD algorithm, which constitutes the inner loop of Paige's algorithm. We pre...