soft_impute.py

'''
   Copyright 2015 Travis Brady

   Licensed under the Apache License, Version 2.0 (the "License");
   you may not use this file except in compliance with the License.
   You may obtain a copy of the License at

       http://www.apache.org/licenses/LICENSE-2.0

   Unless required by applicable law or agreed to in writing, software
   distributed under the License is distributed on an "AS IS" BASIS,
   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   See the License for the specific language governing permissions and
   limitations under the License.
'''
from __future__ import print_function
import numpy as np


def frob(Uold, Dsqold, Vold, U, Dsq, V):
    denom = (Dsqold ** 2).sum()
    utu = Dsq * (U.T.dot(Uold))
    vtv = Dsqold * (Vold.T.dot(V))
    uvprod = utu.dot(vtv).diagonal().sum()
    num = denom + (Dsqold ** 2).sum() - 2*uvprod
    return num / max(denom, 1e-9)

class SoftImpute:
    def __init__(self, J=2, thresh=1e-05, lambda_=0, maxit=100, random_state=None, verbose=False):
        self.J = J
        self.thresh = thresh
        self.lambda_ = lambda_
        self.maxit = maxit
        self.rs = np.random.RandomState(random_state)
        self.verbose = verbose
        self.u = None
        self.d = None
        self.v = None

    def fit(self, X):
        n,m = X.shape
        xnas = np.isnan(X)
        nz = m*n - xnas.sum()
        xfill = X.copy()
        V = np.zeros((m, self.J))
        U = self.rs.normal(0.0, 1.0, (n, self.J))
        U, _, _ = np.linalg.svd(U, full_matrices=False)
        Dsq = np.ones((self.J, 1))
        #xfill[xnas] = 0.0
        col_means = np.nanmean(xfill, axis=0)
        np.copyto(xfill, col_means, where=np.isnan(xfill))
        ratio = 1.0
        iters = 0
        while ratio > self.thresh and iters < self.maxit:
            iters += 1
            U_old = U
            V_old = V
            Dsq_old = Dsq
            B = U.T.dot(xfill)

            if self.lambda_ > 0:
                tmp = (Dsq / (Dsq + self.lambda_))
                B = B * tmp

            Bsvd = np.linalg.svd(B.T, full_matrices=False)
            V = Bsvd[0]
            Dsq = Bsvd[1][:, np.newaxis]
            U = U.dot(Bsvd[2])

            tmp = Dsq * V.T

            xhat = U.dot(tmp)

            xfill[xnas] = xhat[xnas]
            A = xfill.dot(V).T
            Asvd = np.linalg.svd(A.T, full_matrices=False)
            U = Asvd[0]
            Dsq = Asvd[1][:, np.newaxis]
            V = V.dot(Asvd[2])
            tmp = Dsq * V.T

            xhat = U.dot(tmp)
            xfill[xnas] = xhat[xnas]
            ratio = frob(U_old, Dsq_old, V_old, U, Dsq, V)
            if self.verbose:
                print('iter: %4d ratio = %.5f' % (iters, ratio))

        self.u = U[:,:self.J]
        self.d = Dsq[:self.J]
        self.v = V[:,:self.J]
        return self

    def suv(self, vd):
        res = self.u.dot(vd.T)
        return res

    def predict(self, X, copyto=False):
        vd = self.v * np.outer(np.ones(self.v.shape[0]), self.d)
        X_imp = self.suv(vd)
        if copyto:
            np.copyto(X, X_imp, where=np.isnan(X))
        else:
            return X_imp

def main():
    X = np.random.random((10,3)) + (np.arange(10).reshape(10,1) ** 2)

    clf = SoftImpute(J=2, lambda_=0.0)
    fit = clf.fit(X)
    X_test = X.copy()
    X_test[3,1] = np.nan
    X_imp = clf.predict(X_test)

if __name__ == '__main__':
    main()