Add a step-by-step PALM4MSA test python script.

misc/test/src/Python/step-by-step_palm4msa.py

+from pyfaust.fact import palm4msa
+from pyfaust.factparams import ParamsPalm4MSA, StoppingCriterion
+from pyfaust.proj import splin, spcol
+import pyfaust
+from numpy.linalg import norm
+import numpy as np
+
+num_its = 100 # the total number of iterations to run PALM4MSA
+
+# Generate a matrix to factorize
+d = 64
+S = pyfaust.rand(d, d, num_factors=2, density=0.1, per_row=True)
+M = S.toarray()
+
+# Set the set of parameters for PAL4MSA
+projs = [splin((d,d), 10), spcol((d,d), 5)]
+
+# set just one iteration of the step-by-step mode
+stop_crit = StoppingCriterion(num_its=1)
+
+# pack all these parameters into a ParamsPalm4MSA
+p = ParamsPalm4MSA(projs, stop_crit)
+
+rel_errs = [] # relative errors for all iterations
+
+# Runs one iteration at a time and compute the relative error
+for i in range(num_its):
+    F, scale = palm4msa(M, p, ret_lambda=True)
+    # backup the error
+    rel_errs += [(F-M).norm()/np.linalg.norm(M)]
+    # retrieve the factors from the Faust F obtained after the one-iteration execution
+    # it's needed to convert them explicitely as Fortran (that is column major
+    # order) numpy array, otherwise it would fail the next iteration
+    facts = [np.asfortranarray(F.factors(0).toarray()),
+             np.asfortranarray(F.factors(1).toarray())]
+    # F is multiplied by lambda, we need to undo that before continuing to the
+    # next iteration, the factor which was multiplied by scale can be any of
+    # them but because all the others are normalized, it's easy to deduce which
+    # one it is
+    # NOTE: remember we want to set the factors exactly as they were at the end of
+    # iteration 0, that's why all must be normalized (by dividind by scale)
+    if norm(facts[0])  > norm(facts[1]):
+        facts[0] /= scale
+    else:
+        facts[1] /= scale
+        # all is ready to pack the parameters and initial state (the factors and
+        # the scale) in a new ParamsPalm4MSA instance
+        p = ParamsPalm4MSA(projs, stop_crit,
+                           init_facts=facts,
+                           init_lambda=scale)
+
+
+print("relative errors along the iterations (step-by-step mode):", rel_errs)
+
+
+stop_crit = StoppingCriterion(num_its=num_its)
+p = ParamsPalm4MSA(projs, stop_crit)
+G = palm4msa(M, p)
+print("Relative error when running all PALM4MSA iterations at once: ", (G-M).norm() / norm(M))
+print("Last relative error obtained when running PALM4MSA "
+      "iteration-by-iteration: ", rel_errs[-1])
+print("Relative error comparing the final Fausts obtained either by in"
+      " step-by-step PALM4MSA versus all-iterations-at-once PALM4MSA: ", (F-G).norm() / G.norm())