The Restricted Isometry Property for Banded Random Matrices

Castorena, Juan; Creusere, Charles D.

doi:10.1109/tsp.2014.2345350

Cited by 11 publications

(10 citation statements)

References 47 publications

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“…For BDM, Theorem 1 can be extended trivially to any sparsity basis, with performance depending on the incoherence of the basis with the canonical basis (Fourier has optimal incoherence). While general BRM results have not been derived [13], it is likely that similar general results could be shown.…”

Section: Analytic Recovery Guaranteementioning

confidence: 84%

“…Second, note that the statement of the theorem doesn't explicitly depend on the degree of localization L or the matrix type. While the recovery guarantee doesn't depend on L, it is possible that the embedding quality (reflected in the isometry constant δ or scaling constants) could vary with L. While we have unified the result for simple exposition, the result is not tight for BRMs [13] and polylog factors could be reduced. Third, while we have presented the results above to keep the exposition as simple as possible, the theorem above could be generalized to characterize the uniqueness and stablilty of the recovery with respect to measurement and modeling error [27].…”

Section: Analytic Recovery Guaranteementioning

confidence: 91%

“…In cases where the matrix Φ is structured, such recovery guarantees often require slightly more measurements (i.e., polylog in N ) and they are no longer universal because the performance changes depending on Ψ [41,31,2,64]. Of special interest for this paper are recent results detailing the properties of random Block Diagonal Matrices (BDM) [52,23] and Banded Random Matrices (BRM) [13], which are structured sparse matrices requiring only local data aggregation for compression. In the context of sparse coding, this means that the codes inferred for signals compressed with certain matrices will almost surely be equivalent to codes inferred for the uncompressed signals.…”

Section: Randomized Linear Dimensionality Reductionmentioning

confidence: 99%

“…The BDM represents a group of RGCs receiving input from a group of photoreceptors, capturing local wiring in a structure that admits strong analysis but potentially exaggerating the partitioning of RGCs into subgroups. The second is a banded random matrix (BRM), composed of independently randomized rows with a compact support that shifts (with overlap) on each row [13]. The BRM represents a model where each subsequent RGC receives input from a limited set of photoreceptors that has partial overlap with the photoreceptors driving neighboring RGCs.…”

Section: Biologically Plausible Local Wiring Constraints In Random Mamentioning

confidence: 99%

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Learning sparse codes from compressed representations with biologically plausible local wiring constraints

Fallah

Willats

Liu

et al. 2020

Preprint

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Sparse coding is an important method for unsupervised learning of task-independent features in theoretical neuroscience models of neural coding. While a number of algorithms exist to learn these representations from the statistics of a dataset, they largely ignore the information bottlenecks present in fiber pathways connecting cortical areas. For example, the visual pathway has many fewer neurons transmitting visual information to cortex than the number of photoreceptors. Both empirical and analytic results have recently shown that sparse representations can be learned effectively after performing dimensionality reduction with randomized linear op- erators, producing latent coefficients that preserve information. Unfortunately, current proposals for sparse coding in the compressed space require a centralized compression process (i.e., dense random matrix) that is biologically unrealistic due to local wiring constraints observed in neural circuits. The main contribution of this paper is to leverage recent results on structured random matrices to propose a theoretical neuroscience model of randomized projections for communication be- tween cortical areas that is consistent with the local wiring constraints observed in neuroanatomy. We show analytically and empirically that unsupervised learning of sparse representations can be performed in the compressed space despite significant local wiring constraints in compression matrices of varying forms (corresponding to different local wiring patterns). Our analysis verifies that even with significant local wiring constraints, the learned representations remain qualitatively similar, have similar quantitative performance in both training and generalization error, and are consistent across many measures with measured macaque V1 receptive fields.

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Section: Analytic Recovery Guaranteementioning

confidence: 84%

Section: Analytic Recovery Guaranteementioning

confidence: 91%

Section: Randomized Linear Dimensionality Reductionmentioning

confidence: 99%

Section: Biologically Plausible Local Wiring Constraints In Random Mamentioning

confidence: 99%

See 2 more Smart Citations

Learning sparse codes from compressed representations with biologically plausible local wiring constraints

Fallah

Willats

Liu

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Both RIP [16][17][18][19] and coherence [9][10][11][12][13][20][21][22][23][24][25][26][27] are important tools to analyze the property of measurement matrices. In this paper, coherence will be adopted to analyze and illustrate the property of constructed measurement matrices, because it is easier to compute.…”

Section: Lemma 11mentioning

confidence: 99%

Binary Sequence Family for Chaotic Compressed Sensing

2019

KSII TIIS

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It is significant to construct deterministic measurement matrices with easy hardware implementation, good sensing performance and good cryptographic property for practical compressed sensing (CS) applications. In this paper, a deterministic construction method of bipolar chaotic measurement matrices is presented based on binary sequence family (BSF) and Chebyshev chaotic sequence. The column vectors of these matrices are the sequences of BSF, where 1 is substituted with-1 and 0 is with 1. The proposed matrices, which exploit the pseudo-randomness of Chebyshev sequence, are sensitive to the initial state. The performance of proposed matrices is analyzed from the perspective of coherence. Theoretical analysis and simulation experiments show that the proposed matrices have limited influence on the recovery accuracy in different initial states and they outperform their Gaussian and Bernoulli counterparts in recovery accuracy. The proposed matrices can make the hardware implement easy by means of linear feedback shift register (LFSR) structures and numeric converter, which is conducive to practical CS.

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