How to solve the following minimization problem: $$\min_{S>0}{F(\mathbf{S}) }= \frac{1}{2}\Vert \mathbf{M} - \mathbf{K_2SK_1^T}\Vert _F^2+\frac{1}{20}\Vert\mathbf{S}\Vert_F^2$$ where $\mathbf{S}\in R^{256 \times 256}$ with nonegative elements, $\mathbf{M}\in R^{n \times m}$, $\mathbf{K_2} \in R^{n \times 256}$, $\mathbf{K_1} \in R^{m \times 256}$. In most cases $3500\lt m \lt 18000$, $8 \lt n \lt 128$.

The data of a minimal case can be downloaded here. In this case $m=3788$, $n=16$. The following code help to load the data into workspace:



import scipy.io
data = scipy.io.loadmat('/home/ubuntu/MATLAB/problem.mat')
K1 = data['K1']
K2 = data['K2']
M = data['M']
S_inital_guess = data['S00']

What I've tried

  1. Vectorize the problem using $\mathbf{K}=kron(\mathbf{K_2},\mathbf{K_1})$. But $\mathbf{K}$ is too large for ordinary PC. And any optimization strategy using hessian matrix would produce more larger matrice.

  2. Solving the matrix-form problem directly which produce a 4-order Hessian tesnsor. Without hession, the algorithm(steepest descent with exact/inexact line search) converges too slowly.

  3. CVXPY - out of memory

    n = 256

    X = cp.Variable((n,n))

    constraints = [X>=0]

    gamma = cp.Parameter(nonneg=True, value=1)

    obj = cp.Minimize(cp.norm(K2 @ X @ K1.transpose() - M,'fro') + gamma*cp.norm(X,'fro')**2)

    prob = cp.Problem(obj,constraints)


How to solve it?

How to solve this large scale minimization problem efficiently? Could you please give me some code (python or matlab) snippet to solve the attach problem? Are there any out-of-box toolboxes I could use?

For Algorithm Testing

I've added a new mat file containing $K_1$,$K_2$,$M$ and a right answer $Xtrue$ for testing. All matrix are much smaller than the original problem in this file.

  • $\begingroup$ Isn't this problem convex? Have you tried cvxpy? cvxpy.org/examples/index.html $\endgroup$
    – snar
    Sep 9 '20 at 2:08
  • $\begingroup$ @snar All these packages only implemented vector based optimization algorithms. If I vectorize my probelm, there will be a memory error(out of memory) when combine $K_1_2=kron(K_2,K_1)$. File "<ipython-input-18-ab7485ffcf93>", line 17, in <module> K = np.kron(K2,K1) File "/home/ubuntu/anaconda3/lib/python3.7/site-packages/numpy/lib/shape_base.py", line 1059, in kron result = concatenate(result, axis=axis) MemoryError $\endgroup$
    – Shannon
    Sep 10 '20 at 1:33
  • 1
    $\begingroup$ Your matrix are full or sparse? Even if full, your $F$ can take a vector, build a matrix from it and compute your formula. This shouldn't produce a memory error. Regarding the algorithm, maybe conjugate gradient or L-BFGS? L-BFGS is well suited for this kind of situation (many variables, large hessian) $\endgroup$ Sep 10 '20 at 6:09
  • $\begingroup$ If this is too heavy for interior-point method algorithms, you can solve it using first-order methods. These are suitable for very large scale problems. You can either code a gradient-descent algorithm yourself or use an existing solver (but there's still some work for you to do). You can also use an accelerated first order scheme to get faster convergence. $\endgroup$
    – iarbel84
    Sep 10 '20 at 9:56
  • $\begingroup$ @Jean-ClaudeArbaut All matrix are dense. I've tried to use L-BFGS-B (3.0 fortran code provided by the algorithm authors) to solve a smaller case, but it converges too slowly,even slower than steepest descent. I also tried scipy.optimize.minimize(method='L-BFGS-B'), which is also slower than SD. Could you please give me some working code to solve this kind of problems? $\endgroup$
    – Shannon
    Sep 11 '20 at 1:26

Here is a simple Julia script. If you translate it to another language beware of the nested loops. Julia handles these efficiently but they should be vectorized for Matlab or Python.

The first time the script is run it will create tab-separated-values (TSV) files for the $X$ and $W$ matrices. On subsequent runs, the script will read the TSV files, execute $k_{max}$ iterations, update the TSV files, and exit.

Thus you can intermittently refine the solution until you run out of patience.

#!/usr/bin/env  julia

#  Sequential Coordinate-wise algorithm for Non-Negative Least-Squares
#  as described on pages 10-11 of
#     http://users.wfu.edu/plemmons/papers/nonneg.pdf
#  Convergence is painfully slow, but unlike most other NNLS
#  algorithms the objective function is reduced at each step.
#  The algorithm described in the PDF was modified from its
#  original vector form:  |Ax - b|²
#    to the matrix form:  |LXKᵀ - M|²  +  λ|X|²
#  and to include the regularization term.

using LinearAlgebra, MAT, DelimitedFiles

function main()
  matfile = "problem.mat"
  Xfile   = "problem.mat.X.tsv"
  Wfile   = "problem.mat.W.tsv"

# read the matrices from the Matlab file
  f = matopen(matfile)
    K = read(f,"K1"); println("K: size = $(size(K)),\t rank = $(rank(K))")
    L = read(f,"K2"); println("L: size = $(size(L)),\t rank = $(rank(L))")
    M = read(f, "M"); println("M: size = $(size(M)),\t rank = $(rank(M))")
  # S = read(f,"S00");println("S: size = $(size(S)),\t rank = $(rank(S))")

  A = L'L
  B = K'K
  C = -L'M*K
  m,n = size(C)
  λ = 1/10     # regularization parameter
  kmax = 100   # maximum iterations

# specify the size of the work arrays
  X = 0*C
  W = 1*C
  H = A[:,1] * B[:,1]'

# resume from latest saved state ... or reset to initial conditions
     X = readdlm(Xfile);  println("X: size = $(size(X)), extrema = $(extrema(X))")
     W = readdlm(Wfile);  println("W: size = $(size(W)), extrema = $(extrema(W))")
     @warn "Could not read the saved X,W matrices; re-initializing."
     X = 0*C
     W = 1*C

  fxn = (norm(L*X*K' - M)^2 + λ*norm(X)^2) / 2
  println("at step 0, fxn = $fxn")

  k = 0
  while k < kmax
     for i = 1:m
         for j = 1:n
             mul!(H, A[:,i], B[:,j]')
             H[i,j] += λ
             δ = min( X[i,j], W[i,j]/H[i,j] )
             X[i,j] -= δ
             H .*= δ
             W .-= H
     k += 1
     fx2 = (norm(L*X*K' - M)^2 + λ*norm(X)^2) / 2
     println("after step $k, fxn = $fx2")

     # convergence check
     if fx2 ≈ fxn; break; end
     fxn = fx2

# save the current state for the next run
  writedlm(Xfile, X)
  writedlm(Wfile, W)

# peek at the current solution
  println("\nsummary of current solution")
  println(" vector(X) = $(X[1:4]) ... $(X[end-3:end])")
  println("extrema(X) = $(extrema(X))")

# invoke the main function                                           
  • $\begingroup$ This works fine for small scale problems. But for this method, it seems that $\lambda$ has no obvious effect on the results. I've changed $\lambda$ from 1 to 100, but the results has no obvious differences. In general, $\lambda$ is a parameter which controls the importance of the regularization term. The bigger $\lambda$ is, the broader the peaks are. Would you please help me to explain this phenomenon? $\endgroup$
    – Shannon
    Sep 27 '20 at 1:50
  • $\begingroup$ For large scale problem,(e.g. 256 *256), this method would produce discontinous peaks(with sharp edge:drive.google.com/file/d/1lAkg_FQ39MM9BWr60YjRDSpaPuDj0zZr/…), which should not exist in my problem. There has to be only smooth peaks in the results. This why I use a L2-norm term as a regularization term. $\endgroup$
    – Shannon
    Sep 27 '20 at 1:50
  • $\begingroup$ The matrix elements $H_{ij}$ are on the order of $\pm 50000$. In the algorithm, $\lambda$ is used to perturb these elements. In order to have a measurable effect $\lambda$ needs to be comparable, i.e. much larger than $100$. $\endgroup$
    – greg
    Sep 27 '20 at 3:12
  • $\begingroup$ Any ideas to avoid those sharp edges? $\endgroup$
    – Shannon
    Nov 10 '20 at 1:17

You can use the projected gradient method, or an accelerated projected gradient method such as FISTA. It's not too hard to implement these yourself.

We could vectorize $S$ but it's more elegant to work directly in the vector space $V$ of $256 \times 256$ matrices with entries in $\mathbb R$. We'll need to know the gradient of your function $F$.

The gradient of the function $h(S) = \frac{1}{20} \| S \|_F^2$ is $$ \nabla h(S) = \frac{1}{10} S. $$

The gradient of the function $g(S) = \frac12 \| M - K_2 S K_1^T \|_F^2$ requires a bit more effort. Let $A$ be the linear transformation defined by $$ A(S) = K_2 S K_1^T. $$ Then $$\nabla g(S) = A^*(A(S) - M) $$ where $A^*$ is the adjoint of $A$. If we can figure out what the adjoint of $A$ is, we'll be done.

The defining property of $A^*$ is $$ \tag{1} \langle A(S), U \rangle = \langle S, A^*(U) \rangle $$ for all $S, U$. But note that, from the definition of the Frobenius inner product, we have \begin{align} \langle A(S), U \rangle &= \text{Tr}((K_2 S K_1^T)^T U) \\ &= \text{Tr}(K_1 S^T K_2^T U) \\ &= \text{Tr}(S^T K_2^T U K_1 ) \qquad (\text{because Tr}(XY) = \text{Tr}(YX) )\\ &= \langle S, K_2^T U K_1 \rangle \end{align} Comparing this with (1), we see that $$ A^*(U) = K_2^T U K_1. $$

So now we're ready to minimize your function $F$ using the projected gradient iteration, which is $$ S^{k+1} = \max(S^k - t \nabla F(S^k), 0) $$ for $k = 0, 1, \ldots$.

You only need to modify a couple lines of code to implement an accelerated projected gradient method (such as FISTA), which will probably converge dramatically faster.

  • $\begingroup$ Would please give me an example using FISTA to solve Ridge regression probelm? MATLAB,Python,Julia, C/C++ etc.Any languge is okay. I can only find some articles using FISTA to solve LASSO[ceremade.dauphine.fr/~carlier/FISTA]. $\endgroup$
    – Shannon
    Sep 27 '20 at 2:40

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