Abstract
This chapter goes through the fundamental connections between statistical mechanics and estimation theory by focusing on the particular problem of compressive sensing. We first show that the asymptotic analysis of a sparse recovery algorithm is mathematically equivalent to the problem of calculating the free energy of a spin glass in the thermodynamic limit. We then use the replica method from statistical mechanics to evaluate the performance in the asymptotic regime. The asymptotic results have several applications in communications and signal processing. We briefly go through two instances of these applications: Characterization of joint sparse recovery algorithms used in distributed compressive sensing and tuning of receivers employed for detection of spatially modulated signals.
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Notes
- 1.
This is typically the case in classic signal sampling techniques.
- 2.
Note that for random S, the density of state is random.
- 3.
A Haar matrix is a random matrix generated from the rotation-invariant measure on the set of all orthonormal matrices.
- 4.
Another well-known example in compressive sensing is the row-orthonormal random sensing matrix; see [39] and references therein for the exact definition and further examples.
- 5.
By support, we refer to the indices of non-zero entries in a vector.
- 6.
Note that the index j is dropped, as we consider a single-terminal setting.
- 7.
Or to learning algorithms in a Bayesian framework.
- 8.
Note that this is the case for any choice of the regularization function.
- 9.
More precisely, if \(F_j \left ( \lambda \right )\) is different from a step function at a single mass point, i.e., derivative of \(F_j \left ( \lambda \right )\) is different from a Dirac impulse at a single mass point, \(\mbox{R}_j \left ( w \right )\) is strictly increasing; for details, see [59, Appendix E].
- 10.
This follows the same reasons given in Sect. 5.4 for the asymptotic analysis of RLS-based algorithms.
- 11.
In fact, the main two macroscopic parameters of a thermodynamic system are entropy and energy. The free energy is derived by applying the second law of thermodynamics as the Lagrange dual function. We however use directly the free energy in our formulation, for sake of brevity.
- 12.
This means that there is no energy flow.
- 13.
In statistical mechanics, this randomizer is known to have quenched randomness. This is different from the type of randomness considered for the microstate.
- 14.
Here, the expectation is taken over the randomizer Ω.
- 15.
Remember that for some choices of regularization function, e.g., ℓ 0-norm, this problem is not even numerically solvable.
- 16.
Note that the expectation is taken with respect to all random variables.
- 17.
In the notation, we drop the integration set for sake of compactness.
- 18.
There are in general various ways to test the validity. The most common test is the zero-temperature entropy test; see [2]. For computationally feasible approaches, one can compare the given solution with large-dimensional (but still finite-dimensional) simulations; for instance, see the consistency the RS solution with numerical simulations in [2, Chapter 6] for ℓ 1-norm minimization.
- 19.
The invalidity of the solution in these cases is shown by the zero-temperature entropy test. For some particular cases, the RS solution violates the known rigorous bounds.
- 20.
- 21.
Since temporal correlation is usually avoided by classic sampling approaches.
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Bereyhi, A., Müller, R.R., Schulz-Baldes, H. (2022). Analysis of Sparse Recovery Algorithms via the Replica Method. In: Kutyniok, G., Rauhut, H., Kunsch, R.J. (eds) Compressed Sensing in Information Processing. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-09745-4_5
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