Moore-Penrose推广了一个大型稀疏矩阵的逆

Question

我有一个方形矩阵,有几万行和一列,只有几1吨0,所以我用这个Matrix包以高效的方式存储在R中.由于base::matrix内存不足,对象无法处理该数量的单元格.

我的问题是我需要逆矩阵以及此类矩阵的Moore-Penrose广义逆,这是我目前无法计算的.

我尝试过的:

• solve产生Error in LU.dgC(a) : cs_lu(A) failed: near-singular A (or out of memory)错误
• MASS::ginv与Matrix班级不相容
• 没有直接的方法将稀疏转换Matrix为例如bigmemory::big.matrix,而后者也MASS::ginv无论如何也无法工作

• 如果我尝试计算矩阵的Choleski分解以便稍后调用Matrix::chol2inv它,我会收到以下错误消息:

  Error in .local(x, ...) : 
    internal_chm_factor: Cholesky factorization failed
  In addition: Warning message:
  In .local(x, ...) :
    Cholmod warning 'not positive definite' at file ../Cholesky/t_cholmod_rowfac.c, line 431
  

• 基于相关问题,我还尝试pbdDMAT了单个节点上的包,但pbdDMAT::chol产生了Cholmod error 'out of memory' at file ../Core/cholmod_memory.c, line 147错误消息

问题简而言之:有没有办法计算这种稀疏矩阵的逆和Moore-Penrose广义逆,除了回到matrix在具有大量RAM的计算机上使用类？

---

快速重现的例子:

library(Matrix)
n <- 1e5
m <- sparseMatrix(1:n, 1:n, x = 1)
m <- do.call(rBind, lapply(1:10, function(i) {
    set.seed(i)
    m[-sample(1:n, n/3), ]
}))
m <- t(m) %*% m

一些描述(感谢Gabor Grothendieck):

> dim(m)
[1] 100000 100000
> sum(m) / prod(dim(m))
[1] 6.6667e-05
> table(rowSums(m))

    0     1     2     3     4     5     6     7     8     9    10 
    5    28   320  1622  5721 13563 22779 26011 19574  8676  1701 
> table(colSums(m))

    0     1     2     3     4     5     6     7     8     9    10 
    5    28   320  1622  5721 13563 22779 26011 19574  8676  1701 

还有一些错误信息:

> Matrix::solve(m)
Error in LU.dgC(a) : cs_lu(A) failed: near-singular A (or out of memory)
> base::solve(m)
Error in asMethod(object) : 
  Cholmod error 'problem too large' at file ../Core/cholmod_dense.c, line 105
> MASS::ginv(m)
Error in MASS::ginv(m) : 'X' must be a numeric or complex matrix

赏金的目标是找到一种计算Moore-Penrose广义逆的方法,m其中RAM小于8Gb(速度和性能并不重要).

Answer 1

如果你只有很少的1,那么你的矩阵在任何一列和任何一行中可能不会超过1,在这种情况下,广义逆等于转置:

library(Matrix)
set.seed(123)
n <- 1e5
m <- sparseMatrix(sample(1:n, n/10), sample(1:n, n/10), x = 1, dims = c(n, n))

##############################################################################
# confirm that m has no more than one 1 in each row and column
##############################################################################
table(rowSums(m))  # here 90,000 rows are all zero, 10,000 have a single one
##     0     1 
## 90000 10000 

table(colSums(m))  # here 90,000 cols are all zero, 10,000 have a single one
##     0     1 
## 90000 10000 

##############################################################################
# calculate generalized inverse
##############################################################################
minv <- t(m)

##############################################################################
# check that when multiplied by minv it gives a diagonal matrix of 0's and 1's
##############################################################################
mm <- m %*% minv

table(diag(mm)) # diagonal has 90,000 zeros and 10,000 ones
##     0     1 
## 90000 10000 

diag(mm) <- 0
range(mm) # off diagonals are all zero
## [1] 0 0

修订改进的演示文稿.