Reduce dimensionality of DSM by linear projection of row vectors into a lower-dimensional subspace. Various projections methods with different properties are available.
dsm.projection(model, n,
method = c("svd", "rsvd", "asvd", "ri", "ri+svd"),
oversampling = NA, q = 2, rate = .01, power=1,
with.basis = FALSE, verbose = FALSE)
A numeric matrix with n columns (latent dimensions) and the same number of rows as the original DSM. Some SVD-based algorithms may discard poorly conditioned singular values, returning fewer than n columns.
If with.basis=TRUE and an orthogonal projection is used, the corresponding orthogonal basis \(B\) of the latent subspace is returned as an attribute "basis". \(B\) is column-orthogonal, hence \(B^T\) projects into latent coordinates and \(B B^T\) is an orthogonal subspace projection in the original coordinate system.
For orthogonal projections, the attribute "R2" contains a numeric vector specifying the proportion of the squared Frobenius norm of the original matrix captured by each of the latent dimensions. If the original matrix has been centered (so that a SVD projection is equivalent to PCA), this corresponds to the proportion of variance “explained” by each dimension.
For SVD-based projections, the attribute "sigma" contains the singular values corresponding to latent dimensions. It can be used to adjust the power scaling exponent at a later time.
either an object of class dsm, or a dense or sparse numeric matrix
projection method to use for dimensionality reduction (see “DETAILS” below)
an integer specifying the number of target dimensions. Use n=NA to generate as many latent dimensions as possible (i.e. the minimum of the number of rows and columns of the DSM matrix).
oversampling factor for stochastic dimensionality reduction algorithms (rsvd, asvd, ri+svd). If unspecified, the default value is 2 for rsvd, 10 for asvd and 10 for ri+svd (subject to change).
number of power iterations in the randomized SVD algorithm (Halko et al. 2009 recommend q=1 or q=2)
fill rate of random projection vectors. Each random dimension has on average rate * ncol(model) nonzero components in the original space
apply power scaling after SVD-based projection, i.e. multiply each latent dimension with a suitable power of the corresponding singular value.
The default power=1 corresponds to a regular orthogonal projection. For power \(> 1\), the first SVD dimensions -- i.e. those capturing the main patterns of \(M\) -- are given more weight; for power \(< 1\), they are given less weight. The setting power=0 results in a full equalization of the dimensions and is also known as “whitening” in the PCA case.
if TRUE, also returns orthogonal basis of the subspace as attribute of the reduced matrix (not available for random indexing methods)
if TRUE, some methods display progress messages during execution
Stephanie Evert (https://purl.org/stephanie.evert)
The following dimensionality reduction algorithms can be selected with the method argument:
singular value decomposition (SVD), using the efficient SVDLIBC algorithm (Berry 1992) from package sparsesvd if the input is a sparse matrix. If the DSM has been scored with scale="center", this method is equivalent to principal component analysis (PCA).
randomized SVD (Halko et al. 2009, p. 9) based on a factorization of rank oversampling * n with q power iterations.
approximate SVD, which determines latent dimensions from a random sample of matrix rows including oversampling * n data points. This heuristic algorithm is highly inaccurate and has been deprecated.
random indexing (RI), i.e. a projection onto random basis vectors that are approximately orthogonal. Basis vectors are generated by setting a proportion of rate elements randomly to \(+1\) or \(-1\). Note that this does not correspond to a proper orthogonal projection, so the resulting coordinates in the reduced space should be used with caution.
RI to oversampling * n dimensions, followed by SVD of the pre-reduced matrix to the final n dimensions. This is not a proper orthogonal projection because the RI basis vectors in the first step are only approximately orthogonal.
Berry, Michael~W. (1992). Large scale sparse singular value computations. International Journal of Supercomputer Applications, 6, 13--49.
Halko, N., Martinsson, P. G., and Tropp, J. A. (2009). Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions. Technical Report 2009-05, ACM, California Institute of Technology.
rsvd for the implementation of randomized SVD, and sparsesvd for the SVDLIBC wrapper
# 240 English nouns in space with correlated dimensions "own", "buy" and "sell"
M <- DSM_GoodsMatrix[, 1:3]
# SVD projection into 2 latent dimensions
S <- dsm.projection(M, 2, with.basis=TRUE)
100 * attr(S, "R2") # dim 1 captures 86.4% of distances
round(attr(S, "basis"), 3) # dim 1 = commodity, dim 2 = owning vs. buying/selling
S[c("time", "goods", "house"), ] # some latent coordinates
if (FALSE) {
idx <- DSM_GoodsMatrix[, 4] > .85 # only show nouns on "fringe"
plot(S[idx, ], pch=20, col="red", xlab="commodity", ylab="own vs. buy/sell")
text(S[idx, ], rownames(S)[idx], pos=3)
}
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