R: Tensor EVD

Tensor EVD {tensorEVD}

R Documentation

Tensor EVD

Description

Fast eigen value decomposition (EVD) of the Hadamard product of two matrices

Usage

tensorEVD(K1, K2, ID1, ID2, alpha = 1.0,
          EVD1 = NULL, EVD2 = NULL,
          d.min = .Machine$double.eps, 
          make.dimnames = FALSE, verbose = FALSE)

Arguments

`K1`, `K2`	(numeric) Covariance structure matrices
`ID1`	(character/integer) Vector of length n with either names or indices mapping from rows/columns of `K1` into the resulting tensor product
`ID2`	(character/integer) Vector of length n with either names or indices mapping from rows/columns of `K2` into the resulting tensor product
`alpha`	(numeric) Proportion of variance of the tensor product to be explained by the tensor eigenvectors
`EVD1`	(list) (Optional) Eigenvectors and eigenvalues of `K1` as produced by the `eigen` function
`EVD2`	(list) (Optional) Eigenvectors and eigenvalues of `K2` as produced by the `eigen` function
`d.min`	(numeric) Tensor eigenvalue threshold. Default is a numeric zero. Only eigenvectors with eigenvalue passing this threshold are returned
`make.dimnames`	`TRUE` or `FALSE` to whether add `rownames` and `colnames` attributes to the output
`verbose`	`TRUE` or `FALSE` to whether show progress

Details

Let the n × n matrix K to be the Hadamard product (aka element-wise or entry-wise product) involving two smaller matrices K₁ and K₂ of dimensions n₁ and n₂, respectively,

K = (Z₁ K₁ Z'₁) ⊙ (Z₂ K₂ Z'₂)

where Z₁ and Z₂ are incidence matrices mapping from rows (and columns) of the resulting Hadamard to rows (and columns) of K₁ and K₂, respectively.

Let the eigenvalue decomposition (EVD) of K₁ and K₂ to be K₁ = V₁ D₁ V'₁ and K₂ = V₂ D₂ V'₂. Using properties of the Hadamard and Kronecker products, an EVD of the Hadamard product K can be approximated using the EVD of K₁ and K₂ as

K = V D V'

where D = D₁⊗D₂ is a diagonal matrix containing N = n₁ × n₂ tensor eigenvalues d₁ ≥ ... ≥ d_N ≥ 0 and V = (Z₁⋆Z₂)(V₁⊗V₂) = [v₁,...,v_N] is matrix containing N tensor eigenvectors v_k; here the term Z₁⋆Z₂ is the "face-splitting product" (aka "transposed Khatri–Rao product") of matrices Z₁ and Z₂.

Each tensor eigenvector k is derived separately as a Hadamard product using the corresponding i(k) and j(k) eigenvectors v_1i(k) and v_2j(k) from V₁ and V₂, respectively, this is

v_k = (Z₁v_1i(k))⊙(Z₂v_2j(k))

The tensorEVD function derives each of these eigenvectors v_k by matrix indexing using integer vectors ID1 and ID2. The entries of these vectors are the row (and column) number of K₁ and K₂ that are mapped at each row of Z₁ and Z₂, respectively.

Value

Returns a list object that contains the elements:

values: (vector) resulting tensor eigenvalues.
vectors: (matrix) resulting tensor eigenvectors.
totalVar: (numeric) total variance of the tensor matrix product.

Examples

  require(tensorEVD)
  set.seed(195021)
  
  # Generate matrices K1 and K2 of dimensions n1 and n2
  n1 = 10; n2 = 15
  K1 = crossprod(matrix(rnorm(n1*(n1+10)), ncol=n1))
  K2 = crossprod(matrix(rnorm(n2*(n2+10)), ncol=n2))

  # (a) Example 1. Full design (Kronecker product)
  ID1 = rep(seq(n1), each=n2)
  ID2 = rep(seq(n2), times=n1)

  # Direct EVD of the Hadamard product
  K = K1[ID1,ID1]*K2[ID2,ID2]
  EVD0 = eigen(K)

  # Tensor EVD using K1 and K2
  EVD = tensorEVD(K1, K2, ID1, ID2)

  # Eigenvectors and eigenvalues are numerically equal
  all.equal(EVD0$values, EVD$values)
  all.equal(abs(EVD0$vectors), abs(EVD$vectors)) 

  # (b) If a proportion of variance explained is specified, 
  # only the eigenvectors needed to explain such proportion are derived
  alpha = 0.95
  EVD = tensorEVD(K1, K2, ID1, ID2, alpha=alpha)
  dim(EVD$vectors)

  # For the direct EVD
  varexp = cumsum(EVD0$values/sum(EVD0$values))
  index = 1:which.min(abs(varexp-alpha))
  dim(EVD0$vectors[,index])

  # (c) Example 2. Incomplete design (Hadamard product)
  # Eigenvectors and eigenvalues are no longer equivalent
  n = n1*n2   # Sample size n
  ID1 = sample(seq(n1), n, replace=TRUE) # Randomly sample of ID1
  ID2 = sample(seq(n2), n, replace=TRUE) # Randomly sample of ID2

  K = K1[ID1,ID1]*K2[ID2,ID2]
  EVD0 = eigen(K)
  EVD = tensorEVD(K1, K2, ID1, ID2)

  all.equal(EVD0$values, EVD$values)
  all.equal(abs(EVD0$vectors), abs(EVD$vectors)) 
  
  # However, the sum of the eigenvalues is equal to the trace(K)
  c(sum(EVD0$values), sum(EVD$values), sum(diag(K)))

  # And provide the same approximation for K
  K01 = EVD0$vectors%*%diag(EVD0$values)%*%t(EVD0$vectors)
  K02 = EVD$vectors%*%diag(EVD$values)%*%t(EVD$vectors)
  c(all.equal(K,K01), all.equal(K,K02))

  # When n is different from N=n1xn2, both methods provide different 
  # number or eigenvectors/eigenvalues. The eigen function provides 
  # a number of eigenvectors equal to the minimum between n and N
  # for the tensorEVD, this number is always N

  # (d) Sample size n being half of n1 x n2
  n = n1*n2/2
  ID1 = sample(seq(n1), n, replace=TRUE)
  ID2 = sample(seq(n2), n, replace=TRUE)

  K = K1[ID1,ID1]*K2[ID2,ID2]
  EVD0 = eigen(K)
  EVD = tensorEVD(K1, K2, ID1, ID2)

  c(eigen=sum(EVD0$values>1E-10), tensorEVD=sum(EVD$values>1E-10))

  # (e) Sample size n being twice n1 x n2
  n = n1*n2*2
  ID1 = sample(seq(n1), n, replace=TRUE)
  ID2 = sample(seq(n2), n, replace=TRUE)

  K = K1[ID1,ID1]*K2[ID2,ID2]
  EVD0 = eigen(K)
  EVD = tensorEVD(K1, K2, ID1, ID2)

  c(eigen=sum(EVD0$values>1E-10), tensorEVD=sum(EVD$values>1E-10))

[Package tensorEVD version 0.1.3 Index]