IEEE Circuits and Systems Magazine - Q1 2018 - 37

This problem is central to machine learning and big-data processing. Examples of applications include model
selection in regression/classification, dictionary learning, matrix completion in recommendation systems, image restoration, graphical modelling, natural language
processing, resource management in sensor networks,
and compressive sensing [36], [63]-[65]. It is often the
case that we can cast sparse learning as an optimization problem that involves sparsity-inducing regularizers, such as the , 1 norm, mixed , 1 and , 2 norms, and
the nuclear norm [36]. In this section, we focus on the
problem of robust compressive sensing (CS), which
recovers sparse signals from noisy observations [66].

We remark that CS yields a problem formulation similar
to LASSO [67], sparse coding [24] and sensor selection
problems [68]. Previous research efforts [66], [69]-[74],
focused on software-based approaches for sparse signal recovery, with the support of CPUs/GPUs. Here we
discuss approaches to employ memristor crossbars to
design CS solvers.
A. Preliminaries on CS
Let z * ! R p be a sparse or compressible vector, e.g., a
digital signal or image, to be recovered. We have access
to measurements h = Hz * + v, where q % p, H ! R q # p is
a given measurement matrix, such as a random Gaussian

4
Dimension 100
Dimension 600
Dimension 1,000

0.5

Error Compared to
Interior-Point Solution (%)

Error Compared to
Interior-Point Solution (%)

0.6

0.4
0.3
0.2
0.1
0

3.5
3
2.5
2
1.5
1
0.5

0

1

2

3
4
5
6
7
8
Hardware Variation (%)
(a)

9

0

10

0

1

2

3
4
5
6
7
8
Hardware Variation (%)
(b)

9

10

Figure 6. solution accuracy versus level of hardware variations for different problem sizes n ! {100, 600, 1000}. (a) memristorbased Lp solver. (b) memristor-based qp solver with the same legend as (a).

10

3,000

× 104

Number of ADMM Iteration

Number of ADMM Iteration

9
2,500
2,000
1,500
1,000
500
0

8
7
6
5
4
3
2
1

0.1

1
10
ADMM Parameter ρ
(a)

100

0.1

1
10
ADMM Parameter ρ
(b)

100

Figure 7. number of admm iterations to obtain an e-accuracy solution for different values of admm parameter t under 10%
hardware variation. (a) memristor-based Lp solver; (b) memristor-based qp solver.

first quarter 2018

ieee circuits and systems magazine

37



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