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Random subsets of structured deterministic frames have MANOVA spectra.

ABSTRACT: We draw a random subset of [Formula: see text] rows from a frame with [Formula: see text] rows (vectors) and [Formula: see text] columns (dimensions), where [Formula: see text] and [Formula: see text] are proportional to [Formula: see text] For a variety of important deterministic equiangular tight frames (ETFs) and tight non-ETFs, we consider the distribution of singular values of the [Formula: see text]-subset matrix. We observe that, for large [Formula: see text], they can be precisely described by a known probability distribution-Wachter's MANOVA (multivariate ANOVA) spectral distribution, a phenomenon that was previously known only for two types of random frames. In terms of convergence to this limit, the [Formula: see text]-subset matrix from all of these frames is shown to be empirically indistinguishable from the classical MANOVA (Jacobi) random matrix ensemble. Thus, empirically, the MANOVA ensemble offers a universal description of the spectra of randomly selected [Formula: see text] subframes, even those taken from deterministic frames. The same universality phenomena is shown to hold for notable random frames as well. This description enables exact calculations of properties of solutions for systems of linear equations based on a random choice of [Formula: see text] frame vectors of [Formula: see text] possible vectors and has a variety of implications for erasure coding, compressed sensing, and sparse recovery. When the aspect ratio [Formula: see text] is small, the MANOVA spectrum tends to the well-known Mar?enko-Pastur distribution of the singular values of a Gaussian matrix, in agreement with previous work on highly redundant frames. Our results are empirical, but they are exhaustive, precise, and fully reproducible.

SUBMITTER: Haikin M

PROVIDER: S-EPMC5495238 | biostudies-literature | 2017 Jun

REPOSITORIES: biostudies-literature

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Random subsets of structured deterministic frames have MANOVA spectra.

Haikin Marina M Zamir Ram R Gavish Matan M

Proceedings of the National Academy of Sciences of the United States of America 20170613 26

We draw a random subset of [Formula: see text] rows from a frame with [Formula: see text] rows (vectors) and [Formula: see text] columns (dimensions), where [Formula: see text] and [Formula: see text] are proportional to [Formula: see text] For a variety of important deterministic equiangular tight frames (ETFs) and tight non-ETFs, we consider the distribution of singular values of the [Formula: see text]-subset matrix. We observe that, for large [Formula: see text], they can be precisely descri ...[more]

PMID: 28611224

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Project description:BackgroundWhen assessing connectivity, it is crucial to rely on accurate modeling frameworks that consider species movement preferences and patterns. One important aspect is the level of randomness or unpredictability in the route selection. In this respect, traditional approaches (based on least-cost path or circuit theory) consider species movements unrealistically as totally deterministic or as totally random. A recent approach (randomized shortest path) advocates for choosing intermediate levels of randomness through a single parameter. This parameter may be optimized by validating connectivity surfaces developed from different levels of randomness against observed movement data. However, connectivity models are seldom validated, and it is still unclear how to approach this task. To address this knowledge gap, this paper aims at comparing different validation methods to infer the optimal randomness level in connectivity studies. Additionally, we aimed to disentangle the practical consequences of applying traditional connectivity approaches versus using an optimized level of movement randomness when delineating corridors.MethodsThese objectives were accomplished through the study case of the Iberian lynx, an endangered species whose maintenance and recovery depend on the current connectivity among its population nuclei. We firstly determined a conductance surface based on point selection functions accounting for the behavioral state (territorial or exploratory) of individuals. Secondly, we identified the level of randomness that better fits lynxes' movements with independent GPS locations and different validation techniques. Lastly, we delineated corridors between lynx population nuclei through a) the randomized shortest path approach and the extreme and optimal levels of randomness of each validation method, and b) the traditional connectivity approaches.ResultsAccording to all used validation methodologies, models with intermediate levels of randomness outperformed those with extreme randomness levels representing totally deterministic or random movements. We found differences in the optimal randomness level among validation methods but similar results in the delineation of corridors. Our results also revealed that models with extreme randomness levels (deterministic and random walk) of the randomized path approach provided equivalent corridor networks to those from traditional approaches. Moreover, these corridor networks calculated with traditional approaches showed notable differences in patterns from the corridor network calculated with an optimized randomness level.ConclusionsHere we presented a connectivity model with a solid biological basis that calibrates the level of movement randomness and is supported by comprehensive validation methods. It is thus a step forward in the search and evaluation of connectivity approaches that lead to improved, efficient, and successful management actions.

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Random subsets of structured deterministic frames have MANOVA spectra.

Publications

Random subsets of structured deterministic frames have MANOVA spectra.

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