The columnwise Khatri–Rao product of two matrices is an important matrix type, reprising its role as a structured sensing matrix in many fundamental linear inverse problems. Robust signal recovery in such inverse problems is often contingent on proving the restricted isometry property (RIP) of a certain system matrix expressible as a Khatri–Rao product of two matrices. In this paper, we analyze the RIP of a generic columnwise Khatri–Rao product matrix by deriving two upper bounds for its @math th order restricted isometry constant ( @math -RIC) for different values of @math . The first RIC bound is computed in terms of the individual RICs of the real-valued input matrices participating in the Khatri–Rao product. The second RIC bound is probabilistic and is specified in terms of the input matrix dimensions. We show that the Khatri–Rao product of a pair of @math sized random matrices comprising independent and identically distributed sub-Gaussian entries satisfies @math -RIP with arbitrarily high probability, provided @math exceeds @math . This is a substantially milder condition compared to @math rows needed to guarantee @math -RIP of the input sub-Gaussian random matrices participating in the Khatri–Rao product. Our RIC bounds confirm that the Khatri–Rao product exhibits stronger restricted isometry compared to its constituent matrices for the same RIP order. The proposed RIC bounds are potentially useful in obtaining improved performance guarantees in several sparse signal recovery and tensor decomposition problems.
In @cite_22 , the isometry property of the Kronecker product @math is analyzed in the @math -norm sense for the restricted class of input vectors expressible as vectorized @math -distributed sparse matrices, wherein, @math and @math are the adjacency matrices of two independent uniformly random @math -left regular bipartite graphs. In this paper, we instead analyze the restricted isometry of the columnwise Khatri-Rao product @math , which is equivalent to the RIP of the Kronecker product @math with respect to vectorized sparse diagonal matrices. In our work, we assume that input matrices @math and @math are random with independent subgaussian elements.
@cite_22
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Consider an @math matrix @math with the restricted isometry property of order k and level @math ; that is, the norm of any k-sparse vector in @math is preserved to within a multiplicative factor of @math under application of @math . We show that by randomizing the column signs of such a matrix @math , the resulting map with high probability embeds any fixed set of @math points in @math into @math without distorting the norm of any point in the set by more than a factor of @math . Consequently, matrices with the restricted isometry property and with randomized column signs provide optimal Johnson–Lindenstrauss embeddings up to logarithmic factors in N. In particular, our results improve the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices; for partial Fourier and partial Hadamard matrices, we improve the recent bound @math given by Ailon and Liberty to $m... In this note will consider sufficient conditions for sparse recovery such as Spark, coherence, restricted isometry property (RIP) and null space property (NSP). Then we will discuss the solution of underdetermined linear equations when the matrix is the Kronecker product of matrices. Specially we will explain how NSP behave in the case where the matrix is the Kronecker product of matrices. Let @math be an @math Fourier matrix over @math for some prime @math . We improve upon known lower bounds for the number of rows of @math that must be sampled so that the resulting matrix @math satisfies the restricted isometry property for @math -sparse vectors. This property states that @math is approximately @math for all @math -sparse vectors @math . In particular, if @math , we show that @math rows must be sampled to satisfy the restricted isometry property with constant probability. Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary @math . This problem is now understood to be well-posed and efficiently solvable under suitable assumptions on the measurements and dictionary if the number of measurements scales roughly with the sparsity level. One sufficient condition for such is the @math -restricted isometry property ( @math -RIP), which asks that the sampling matrix approximately preserve the norm of all signals which are sufficiently sparse in @math . While many classes of random matrices are known to satisfy such conditions, such matrices are not representative of the structural constraints imposed by practical sensing systems. We close this gap in the theory by demonstrating that one can subsample a fixed orthogonal matrix in such a way that the @math -RIP will hold, provided this basis is sufficiently incoherent with the sparsifying dictionary $ ...