Schatten Norms in Matrix Streams: Hello Sparsity, Goodbye Dimension
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The spectrum of a matrix contains important structural information about the underlying data, and hence there is considerable interest in computing various functions of the matrix spectrum. A fundamental example for such functions is the l_p-norm of the spectrum, called the Schatten p-norm of the matrix. Large matrices representing real-world data are often sparse (most entries are zeros) or doubly sparse, i.e., sparse in both rows and columns. These large matrices are usually accessed as a stream of updates, typically organized in row-order. In this setting, where space (memory) is the limiting resource, computing spectral functions is an expensive task and known algorithms require space that is polynomial in the dimension of the matrix, even for sparse matrices. Thus, it is highly desirable to design algorithms requiring significantly smaller space. We answer this challenge by providing the first algorithm that uses space independent of the matrix dimension to compute the Schatten p-norm of a doubly-sparse matrix presented in row order. Instead, our algorithm uses space polynomial in the sparsity parameter k and makes O(p) passes over the data stream. We further prove that multiple passes are unavoidable in this setting and show several extensions of our primary technique, including stronger upper bounds for special matrix families, algorithms for the more difficult turnstile model, and a trade-off between space requirements and number of passes.
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