The major challenge of customizing power iteration for memristor implementation is to determine the multiplicity of the dominant eigenvalue and to find the corresponding eigenvectors. pursuit (OMP) algorithm [77], a commonly used software-based CS solver. We observe that the recovery accuracy improves as the signal becomes sparser, namely, s is smaller. This is not surprising, since a sparser signal can be more stably recovered at the rate much smaller than what is commonly prescribed by Shannon-Nyquist theorem [70]. By fixing s, we observe that the recovery accuracy decreases while increasing the level of hardware variations. Although the presence of hardware variations negatively affects the recovery accuracy, the sparse pattern error shown by Figs. 8(b) and (c) is acceptable, as it is below 6%. In particular, in Fig. 8(c) the recovered signal yields almost the same sparse support as that of the original signal even in the presence of 10% hardware variation. These promising results show that the memristor-based CS solver is quite robust to hardware variations, and is able to provide reliable recovered sparse patterns. Lastly, we investigate the convergence of the memristor-based approach against different values of the ADMM parameter t. Similar to Fig. 7, a moderate choice of t, namely, t = 10 in this example, is preferred over others as shown in Fig. 8(d). VI. Power Iteration via Memristors: Application to PCA Principal component analysis (PCA) is the best-known dimensionality-reduction technique to find intrinsic lowdimensional manifolds from high-dimensional data [40]. The implementation of PCA requires the computation of the principal eigenvalues and the corresponding eigenvectors of a symmetric matrix. The calculation of eigenvalues and eigenvectors is also motivated by optimization problems, e.g., a projection onto semidefinite cones in semidefinite programming [78]. Since power iteration (PI) is a widely-used algorithm for eigenvalue analysis [79], here we describe a memristor-based PI framework. A. Preliminaries on PI PI is an iterative algorithm that converges to the eigenvector associated with the largest eigenvalue of a matrix. Let {(m i, u i)} in= 1 denote a set of eigenvalue-eigenvector pairs for matrix A ! R n # n, where we refer to m 1, regardless of its multiplicity, as the dominant eigenvalue. The kth iteration of PI is given by [42] k -1 x k = Axk -1 , < Ax < 2 40 ieee circuits and systems magazine where x 0 is an arbitrary starting vector. If k " 3, then by (32), x k converges to the eigenvector u 1, and thus (x k) T Ax k / (x k) T x k converges to the largest eigenvalue m 1 . The convergence of PI is geometric, with ratio |m 2|/|m 1| [42]. Therefore, PI converges slowly if there is an eigenvalue close in magnitude to the dominant eigenvalue. Moreover, if the largest eigenvalue is not unique, say m 1 = m 2 with multiplicity 2, the limiting point x k fails to converge to u 1, and instead converges to a linear combination of eigenvectors u 1 and u 2 [80]. Thus, it is required that the memristor-based PI be able to address the issue of repeated eigenvalues. B. Memristor-Based PI It is clear from (32) that the PI algorithm involves a) matrix-vector multiplication Ax k -1, and b) evaluation of a vector norm. Based on (2), the first operation is easily implemented using memristor crossbars. And the second operation can be realized using elementary digital (or analog) circuits [30]. The major challenge of customizing PI for memristor implementation is to determine the multiplicity of the dominant eigenvalue and to find the corresponding eigenvectors. In what follows, we show that with the aid of Gram-Schmidt process such a problem can be addressed via elementary matrix-vector operations. We assume that the largest eigenvalue has multiplicity s, namely, m 1 = m 2 = g = m s . Under s random initial vectors, we denote by {y i} is= 1 the converging vectors of PI. It is known from [80] that {y i} is= 1 are linear combinations of eigenvectors {u i} is= 1 . This implies two facts. p First, given p initial vectors, the resulting {y i} i = 1 are linearly independent if p # s. Therefore, we are able to determine the number of repeated dominant eigenvalues by adding new columns to Yp until its rank stops increasing where Yp := [y 1, g, y p], and its rank can be determined by the singularity of Yp Y Tp . Second, given the number of repeated eigenvalues, finding the eigenvectors {u i} is= 1 is equivalent to seeking an orthogonal subspace spanned by {y i} is= 1 . This procedure is precisely described by the Gram-Schmidt process. Given a sequence of vectors {y i} is= 1, the Gram-Schmidt process generates a sequence of orthogonal vectors {u i} is= 1 [42], i -1 ui = yi - / j =1 (32) y Ti u j u j, i = 2, g, s, u Tj u j (33) where u 1 = y 1 . first quarter 2018

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