# Sparse regularized low-rank matrix approximation. Estimate an $l1$-penalized singular value or principal components decomposition (SVD or PCA) that introduces sparsity in the right singular vectors based on the fast and memory-efficient sPCA-rSVD algorithm of Haipeng Shen and Jianhua Huang. ```r ssvd( x, k = 1, n = 2, maxit = 500, tol = 0.001, center = FALSE, scale. = FALSE, alpha = 0, tsvd = NULL, ... ) ``` ## Arguments - `x`: A numeric real- or complex-valued matrix or real-valued sparse matrix. - `k`: Matrix rank of the computed decomposition (see the Details section below). - `n`: Number of nonzero components in the right singular vectors. If `k > 1`, then a single value of `n` specifies the number of nonzero components in each regularized right singular vector. Or, specify a vector of length `k` indicating the number of desired nonzero components in each returned vector. See the examples. - `maxit`: Maximum number of soft-thresholding iterations. - `tol`: Convergence is determined when $||U_j - U_{j-1}||_F < tol$, where $U_j$ is the matrix of estimated left regularized singular vectors at iteration $j$. - `center`: a logical value indicating whether the variables should be shifted to be zero centered. Alternately, a centering vector of length equal the number of columns of `x` can be supplied. Use `center=TRUE` to perform a regularized sparse PCA. - `scale.`: a logical value indicating whether the variables should be scaled to have unit variance before the analysis takes place. Alternatively, a vector of length equal the number of columns of `x` can be supplied. The value of `scale` determines how column scaling is performed (after centering). If `scale` is a numeric vector with length equal to the number of columns of `x`, then each column of `x` is divided by the corresponding value from `scale`. If `scale` is `TRUE` then scaling is done by dividing the (centered) columns of `x` by their standard deviations if `center=TRUE`, and the root mean square otherwise. If `scale` is `FALSE`, no scaling is done. See `scale` for more details. - `alpha`: Optional scalar regularization parameter between zero and one (see Details below). - `tsvd`: Optional initial rank-k truncated SVD or PCA (skips computation if supplied). - `...`: Additional arguments passed to `irlba`. ## Returns A list containing the following components: - **u**: regularized left singular vectors with orthonormal columns - **d**: regularized upper-triangular projection matrix so that `x %*% v == u %*% d` - **v**: regularized, sparse right singular vectors with columns of unit norm - **center**: the centering used, if any - **scale**: the scaling used, if any - **lambda**: the per-column regularization parameter found to obtain the desired sparsity - **iter**: number of soft thresholding iterations - **n**: value of input parameter `n` - **alpha**: value of input parameter `alpha` ## Details The `ssvd` function implements a version of an algorithm by Shen and Huang that computes a penalized SVD or PCA that introduces sparsity in the right singular vectors by solving a penalized least squares problem. The algorithm in the rank 1 case finds vectors $u, w$ that minimize $$ \|x - u w^T\|_F^2 + \lambda \|w\|_1||x - u w^T||_F^2 + lambda||w||_1 $$ such that $||u|| = 1$, and then sets $v = w / ||w||$ and $d = u^T x v$; see the referenced paper for details. The penalty $lambda$ is implicitly determined from the specified desired number of nonzero values `n`. Higher rank output is determined similarly but using a sequence of $lambda$ values determined to maintain the desired number of nonzero elements in each column of `v` specified by `n`. Unlike standard SVD or PCA, the columns of the returned `v` when `k > 1` may not be orthogonal. ## Note Our `ssvd` implementation of the Shen-Huang method makes the following choices: 1. The l1 penalty is the only available penalty function. Other penalties may appear in the future. 2. Given a desired number of nonzero elements in `v`, value(s) for the $lambda$ penalty are determined to achieve the sparsity goal subject to the parameter `alpha`. 3. An experimental block implementation is used for results with rank greater than 1 (when `k \> 1`) instead of the deflation method described in the reference. 4. The choice of a penalty lambda associated with a given number of desired nonzero components is not unique. The `alpha` parameter, a scalar between zero and one, selects any possible value of lambda that produces the desired number of nonzero entries. The default `alpha = 0` selects a penalized solution with largest corresponding value of `d` in the 1-d case. Think of `alpha` as fine-tuning of the penalty. 5. Our method returns an upper-triangular matrix `d` when `k \> 1` so that `x %*% v == u %*% d`. Non-zero elements above the diagonal result from non-orthogonality of the `v` matrix, providing a simple interpretation of cumulative information, or explained variance in the PCA case, via the singular value decomposition of `d %*% t(v)`. What if you have no idea for values of the argument `n` (the desired sparsity)? The reference describes a cross-validation and an ad-hoc approach; neither of which are in the package yet. Both are prohibitively computationally expensive for matrices with a huge number of columns. A future version of this package will include a revised approach to automatically selecting a reasonable sparsity constraint. Compare with the similar but more general functions `SPC` and `PMD` in the `PMA` package by Daniela M. Witten, Robert Tibshirani, Sam Gross, and Balasubramanian Narasimhan. The `PMD` function can compute low-rank regularized matrix decompositions with sparsity penalties on both the `u` and `v` vectors. The `ssvd` function is similar to the PMD(*, L1) method invocation of `PMD` or alternatively the `SPC` function. Although less general than `PMD`(*), the `ssvd` function can be faster and more memory efficient for the basic sparse PCA problem. See [https://bwlewis.github.io/irlba/ssvd.html](https://bwlewis.github.io/irlba/ssvd.html) for more information. (* Note that the s4vd package by Martin Sill and Sebastian Kaiser, [https://cran.r-project.org/package=s4vd](https://cran.r-project.org/package=s4vd), includes a fast optimized version of a closely related algorithm by Shen, Huang, and Marron, that penalizes both `u` and `v`.) ## Examples ```r set.seed(1) u <- matrix(rnorm(200), ncol=1) v <- matrix(c(runif(50, min=0.1), rep(0,250)), ncol=1) u <- u / drop(sqrt(crossprod(u))) v <- v / drop(sqrt(crossprod(v))) x <- u %*% t(v) + 0.001 * matrix(rnorm(200*300), ncol=300) s <- ssvd(x, n=50) table(actual=v[, 1] != 0, estimated=s$v[, 1] != 0) oldpar <- par(mfrow=c(2, 1)) plot(u, cex=2, main="u (black circles), Estimated u (blue discs)") points(s$u, pch=19, col=4) plot(v, cex=2, main="v (black circles), Estimated v (blue discs)") points(s$v, pch=19, col=4) # Let's consider a trivial rank-2 example (k=2) with noise. Like the # last example, we know the exact number of nonzero elements in each # solution vector of the noise-free matrix. Note the application of # different sparsity constraints on each column of the estimated v. # Also, the decomposition is unique only up to sign, which we adjust # for below. set.seed(1) u <- qr.Q(qr(matrix(rnorm(400), ncol=2))) v <- matrix(0, ncol=2, nrow=300) v[sample(300, 15), 1] <- runif(15, min=0.1) v[sample(300, 50), 2] <- runif(50, min=0.1) v <- qr.Q(qr(v)) x <- u %*% (c(2, 1) * t(v)) + .001 * matrix(rnorm(200 * 300), 200) s <- ssvd(x, k=2, n=colSums(v != 0)) # Compare actual and estimated vectors (adjusting for sign): s$u <- sign(u) * abs(s$u) s$v <- sign(v) * abs(s$v) table(actual=v[, 1] != 0, estimated=s$v[, 1] != 0) table(actual=v[, 2] != 0, estimated=s$v[, 2] != 0) plot(v[, 1], cex=2, main="True v1 (black circles), Estimated v1 (blue discs)") points(s$v[, 1], pch=19, col=4) plot(v[, 2], cex=2, main="True v2 (black circles), Estimated v2 (blue discs)") points(s$v[, 2], pch=19, col=4) par(oldpar) ``` ## References * Shen, Haipeng, and Jianhua Z. Huang. "Sparse principal component analysis via regularized low rank matrix approximation." Journal of multivariate analysis 99.6 (2008): 1015-1034. * Witten, Tibshirani and Hastie (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. _Biostatistics_ 10(3): 515-534.

ssvd function