1,810 research outputs found

    Fundamental performance limits for ideal decoders in high-dimensional linear inverse problems

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    This paper focuses on characterizing the fundamental performance limits that can be expected from an ideal decoder given a general model, ie, a general subset of "simple" vectors of interest. First, we extend the so-called notion of instance optimality of a decoder to settings where one only wishes to reconstruct some part of the original high dimensional vector from a low-dimensional observation. This covers practical settings such as medical imaging of a region of interest, or audio source separation when one is only interested in estimating the contribution of a specific instrument to a musical recording. We define instance optimality relatively to a model much beyond the traditional framework of sparse recovery, and characterize the existence of an instance optimal decoder in terms of joint properties of the model and the considered linear operator. Noiseless and noise-robust settings are both considered. We show somewhat surprisingly that the existence of noise-aware instance optimal decoders for all noise levels implies the existence of a noise-blind decoder. A consequence of our results is that for models that are rich enough to contain an orthonormal basis, the existence of an L2/L2 instance optimal decoder is only possible when the linear operator is not substantially dimension-reducing. This covers well-known cases (sparse vectors, low-rank matrices) as well as a number of seemingly new situations (structured sparsity and sparse inverse covariance matrices for instance). We exhibit an operator-dependent norm which, under a model-specific generalization of the Restricted Isometry Property (RIP), always yields a feasible instance optimality property. This norm can be upper bounded by an atomic norm relative to the considered model.Comment: To appear in IEEE Transactions on Information Theor

    Compressible Distributions for High-dimensional Statistics

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    We develop a principled way of identifying probability distributions whose independent and identically distributed (iid) realizations are compressible, i.e., can be well-approximated as sparse. We focus on Gaussian random underdetermined linear regression (GULR) problems, where compressibility is known to ensure the success of estimators exploiting sparse regularization. We prove that many distributions revolving around maximum a posteriori (MAP) interpretation of sparse regularized estimators are in fact incompressible, in the limit of large problem sizes. A highlight is the Laplace distribution and ℓ1\ell^{1} regularized estimators such as the Lasso and Basis Pursuit denoising. To establish this result, we identify non-trivial undersampling regions in GULR where the simple least squares solution almost surely outperforms an oracle sparse solution, when the data is generated from the Laplace distribution. We provide simple rules of thumb to characterize classes of compressible (respectively incompressible) distributions based on their second and fourth moments. Generalized Gaussians and generalized Pareto distributions serve as running examples for concreteness.Comment: Was previously entitled "Compressible priors for high-dimensional statistics"; IEEE Transactions on Information Theory (2012

    Dynamical Systems in Spiking Neuromorphic Hardware

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    Dynamical systems are universal computers. They can perceive stimuli, remember, learn from feedback, plan sequences of actions, and coordinate complex behavioural responses. The Neural Engineering Framework (NEF) provides a general recipe to formulate models of such systems as coupled sets of nonlinear differential equations and compile them onto recurrently connected spiking neural networks – akin to a programming language for spiking models of computation. The Nengo software ecosystem supports the NEF and compiles such models onto neuromorphic hardware. In this thesis, we analyze the theory driving the success of the NEF, and expose several core principles underpinning its correctness, scalability, completeness, robustness, and extensibility. We also derive novel theoretical extensions to the framework that enable it to far more effectively leverage a wide variety of dynamics in digital hardware, and to exploit the device-level physics in analog hardware. At the same time, we propose a novel set of spiking algorithms that recruit an optimal nonlinear encoding of time, which we call the Delay Network (DN). Backpropagation across stacked layers of DNs dramatically outperforms stacked Long Short-Term Memory (LSTM) networks—a state-of-the-art deep recurrent architecture—in accuracy and training time, on a continuous-time memory task, and a chaotic time-series prediction benchmark. The basic component of this network is shown to function on state-of-the-art spiking neuromorphic hardware including Braindrop and Loihi. This implementation approaches the energy-efficiency of the human brain in the former case, and the precision of conventional computation in the latter case

    Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance

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    We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, distance scaling as nÏ” for Ï”>0 and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate k/n of 13/72=0.180
 and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime
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