The Experts below are selected from a list of 87 Experts worldwide ranked by ideXlab platform

Justin Romberg - One of the best experts on this subject based on the ideXlab platform.

  • Blind Deconvolution Using Convex Programming
    IEEE Transactions on Information Theory, 2014
    Co-Authors: Ali Ahmed, Benjamin Recht, Justin Romberg
    Abstract:

    We consider the problem of recovering two unknown vectors, w and x, of length L from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension N and the other with dimension K. Although the observed convolution is nonlinear in both w and x, it is linear in the rank-1 matrix formed by their outer product wx*. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that, for “generic” signals, the program can deconvolve w and x exactly when the maximum of N and K is almost on the order of L. That is, we show that if x is drawn from a random subspace of dimension N, and w is a vector in a subspace of dimension K whose basis vectors are spread out in the frequency domain, then nuclear norm minimization recovers wx* without error. We discuss this result in the context of blind channel estimation in communications. If we have a Message of length N, which we code using a random L x N coding matrix, and the Encoded Message travels through an unknown linear time-invariant channel of maximum length K, then the receiver can recover both the channel response and the Message when L ≳ N + K, to within constant and log factors.

  • blind deconvolution using convex programming
    arXiv: Information Theory, 2012
    Co-Authors: Ali Ahmed, Benjamin Recht, Justin Romberg
    Abstract:

    We consider the problem of recovering two unknown vectors, $\w$ and $\x$, of length $L$ from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension $N$ and the other with dimension $K$. Although the observed convolution is nonlinear in both $\w$ and $\x$, it is linear in the rank-1 matrix formed by their outer product $\w\x^*$. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that for "generic" signals, the program can deconvolve $\w$ and $\x$ exactly when the maximum of $N$ and $K$ is almost on the order of $L$. That is, we show that if $\x$ is drawn from a random subspace of dimension $N$, and $\w$ is a vector in a subspace of dimension $K$ whose basis vectors are "spread out" in the frequency domain, then nuclear norm minimization recovers $\w\x^*$ without error. We discuss this result in the context of blind channel estimation in communications. If we have a Message of length $N$ which we code using a random $L\times N$ coding matrix, and the Encoded Message travels through an unknown linear time-invariant channel of maximum length $K$, then the receiver can recover both the channel response and the Message when $L\gtrsim N+K$, to within constant and log factors.

Jim Schaad - One of the best experts on this subject based on the ideXlab platform.

Ali Ahmed - One of the best experts on this subject based on the ideXlab platform.

  • Blind Deconvolution Using Convex Programming
    IEEE Transactions on Information Theory, 2014
    Co-Authors: Ali Ahmed, Benjamin Recht, Justin Romberg
    Abstract:

    We consider the problem of recovering two unknown vectors, w and x, of length L from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension N and the other with dimension K. Although the observed convolution is nonlinear in both w and x, it is linear in the rank-1 matrix formed by their outer product wx*. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that, for “generic” signals, the program can deconvolve w and x exactly when the maximum of N and K is almost on the order of L. That is, we show that if x is drawn from a random subspace of dimension N, and w is a vector in a subspace of dimension K whose basis vectors are spread out in the frequency domain, then nuclear norm minimization recovers wx* without error. We discuss this result in the context of blind channel estimation in communications. If we have a Message of length N, which we code using a random L x N coding matrix, and the Encoded Message travels through an unknown linear time-invariant channel of maximum length K, then the receiver can recover both the channel response and the Message when L ≳ N + K, to within constant and log factors.

  • blind deconvolution using convex programming
    arXiv: Information Theory, 2012
    Co-Authors: Ali Ahmed, Benjamin Recht, Justin Romberg
    Abstract:

    We consider the problem of recovering two unknown vectors, $\w$ and $\x$, of length $L$ from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension $N$ and the other with dimension $K$. Although the observed convolution is nonlinear in both $\w$ and $\x$, it is linear in the rank-1 matrix formed by their outer product $\w\x^*$. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that for "generic" signals, the program can deconvolve $\w$ and $\x$ exactly when the maximum of $N$ and $K$ is almost on the order of $L$. That is, we show that if $\x$ is drawn from a random subspace of dimension $N$, and $\w$ is a vector in a subspace of dimension $K$ whose basis vectors are "spread out" in the frequency domain, then nuclear norm minimization recovers $\w\x^*$ without error. We discuss this result in the context of blind channel estimation in communications. If we have a Message of length $N$ which we code using a random $L\times N$ coding matrix, and the Encoded Message travels through an unknown linear time-invariant channel of maximum length $K$, then the receiver can recover both the channel response and the Message when $L\gtrsim N+K$, to within constant and log factors.

M Biskup - One of the best experts on this subject based on the ideXlab platform.

  • a word that does not appear in Encoded Message as a resynchronization marker
    Information Theory Workshop, 2008
    Co-Authors: M Biskup
    Abstract:

    In case of variable-length codes a single bit error may cause loss of synchronization at the decoder and thus may lead to error propagation. Even if the decoder resynchronizes after a number of bits, it may have decoded incorrect number of symbols and may place the further decoded symbols at wrong positions. This paper describes a method for choosing such string of bits ws that the decoder can always recognize any insertions of ws into the Encoded Message and reestablish synchronization. ws will be constructed of the shortest word that is not a substring of the Encoded Message. The method does not require any modification of the code.

Benjamin Recht - One of the best experts on this subject based on the ideXlab platform.

  • Blind Deconvolution Using Convex Programming
    IEEE Transactions on Information Theory, 2014
    Co-Authors: Ali Ahmed, Benjamin Recht, Justin Romberg
    Abstract:

    We consider the problem of recovering two unknown vectors, w and x, of length L from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension N and the other with dimension K. Although the observed convolution is nonlinear in both w and x, it is linear in the rank-1 matrix formed by their outer product wx*. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that, for “generic” signals, the program can deconvolve w and x exactly when the maximum of N and K is almost on the order of L. That is, we show that if x is drawn from a random subspace of dimension N, and w is a vector in a subspace of dimension K whose basis vectors are spread out in the frequency domain, then nuclear norm minimization recovers wx* without error. We discuss this result in the context of blind channel estimation in communications. If we have a Message of length N, which we code using a random L x N coding matrix, and the Encoded Message travels through an unknown linear time-invariant channel of maximum length K, then the receiver can recover both the channel response and the Message when L ≳ N + K, to within constant and log factors.

  • blind deconvolution using convex programming
    arXiv: Information Theory, 2012
    Co-Authors: Ali Ahmed, Benjamin Recht, Justin Romberg
    Abstract:

    We consider the problem of recovering two unknown vectors, $\w$ and $\x$, of length $L$ from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension $N$ and the other with dimension $K$. Although the observed convolution is nonlinear in both $\w$ and $\x$, it is linear in the rank-1 matrix formed by their outer product $\w\x^*$. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that for "generic" signals, the program can deconvolve $\w$ and $\x$ exactly when the maximum of $N$ and $K$ is almost on the order of $L$. That is, we show that if $\x$ is drawn from a random subspace of dimension $N$, and $\w$ is a vector in a subspace of dimension $K$ whose basis vectors are "spread out" in the frequency domain, then nuclear norm minimization recovers $\w\x^*$ without error. We discuss this result in the context of blind channel estimation in communications. If we have a Message of length $N$ which we code using a random $L\times N$ coding matrix, and the Encoded Message travels through an unknown linear time-invariant channel of maximum length $K$, then the receiver can recover both the channel response and the Message when $L\gtrsim N+K$, to within constant and log factors.