ensmallen/tests/lrsdp_test.cpp

/**
 * @file lrsdp_test.cpp
 * @author Ryan Curtin
 * @author Marcus Edel
 * @author Conrad Sanderson
 *
 * ensmallen is free software; you may redistribute it and/or modify it under
 * the terms of the 3-clause BSD license.  You should have received a copy of
 * the 3-clause BSD license along with ensmallen.  If not, see
 * http://www.opensource.org/licenses/BSD-3-Clause for more information.
 */

#include <ensmallen.hpp>
#include "catch.hpp"

using namespace arma;
using namespace ens;
using namespace ens::test;

/**
 * Create a Lovasz-Theta initial point.
 */
template<typename MatType>
void CreateLovaszThetaInitialPoint(const MatType& edges,
                                   MatType& coordinates)
{
  // Get the number of vertices in the problem.
  const size_t vertices = max(max(edges)) + 1;

  const size_t m = edges.n_cols + 1;
  float r = 0.5 + sqrt(0.25 + 2 * m);
  if (ceil(r) > vertices)
    r = vertices; // An upper bound on the dimension.

  coordinates.set_size(vertices, ceil(r));

  // Now we set the entries of the initial matrix according to the formula given
  // in Section 4 of Monteiro and Burer.
  for (size_t i = 0; i < vertices; ++i)
  {
    for (size_t j = 0; j < ceil(r); ++j)
    {
      if (i == j)
        coordinates(i, j) = sqrt(1.0 / r) + sqrt(1.0 / (vertices * m));
      else
        coordinates(i, j) = sqrt(1.0 / (vertices * m));
    }
  }
}

/**
 * Prepare an LRSDP object to solve the Lovasz-Theta SDP in the manner detailed
 * in Monteiro + Burer 2004.  The list of edges in the graph must be given; that
 * is all that is necessary to set up the problem.  A matrix which will contain
 * initial point coordinates should be given also.
 */
template<typename MatType>
void SetupLovaszTheta(const MatType& edges,
                      LRSDP<SDP<MatType>>& lovasz)
{
  // Get the number of vertices in the problem.
  const size_t vertices = max(max(edges)) + 1;

  // C = -(e e^T) = -ones().
  lovasz.SDP().C().ones(vertices, vertices);
  lovasz.SDP().C() *= -1;

  // b_0 = 1; else = 0.
  lovasz.SDP().SparseB().zeros(edges.n_cols + 1);
  lovasz.SDP().SparseB()[0] = 1;

  // A_0 = I_n.
  lovasz.SDP().SparseA()[0].eye(vertices, vertices);

  // A_ij only has ones at (i, j) and (j, i) and 0 elsewhere.
  for (size_t i = 0; i < edges.n_cols; ++i)
  {
    lovasz.SDP().SparseA()[i + 1].zeros(vertices, vertices);
    lovasz.SDP().SparseA()[i + 1](edges(0, i), edges(1, i)) = 1.;
    lovasz.SDP().SparseA()[i + 1](edges(1, i), edges(0, i)) = 1.;
  }

  // Set the Lagrange multipliers right.
  lovasz.AugLag().Lambda().ones(edges.n_cols + 1);
  lovasz.AugLag().Lambda() *= -1;
  lovasz.AugLag().Lambda()[0] = -((typename MatType::elem_type) vertices);
}

/**
 * johnson8-4-4.co test case for Lovasz-Theta LRSDP.
 * See Monteiro and Burer 2004.
 */
TEST_CASE("Johnson844LovaszThetaSDP", "[LRSDPTest]")
{
  // Load the edges.
  arma::mat edges;

  if (edges.load("data/johnson8-4-4.csv", arma::csv_ascii) == false)
  {
    FAIL("couldn't load data");
    return;
  }

  edges = edges.t();

  // The LRSDP itself and the initial point.
  arma::mat coordinates;

  CreateLovaszThetaInitialPoint(edges, coordinates);

  LRSDP<SDP<arma::mat>> lovasz(edges.n_cols + 1, 0, coordinates);

  SetupLovaszTheta(edges, lovasz);

  double finalValue = lovasz.Optimize(coordinates);

  // Final value taken from Monteiro + Burer 2004.
  REQUIRE(finalValue == Approx(-14.0).epsilon(1e-7));

  // Now ensure that all the constraints are satisfied.
  arma::mat rrt = coordinates * trans(coordinates);
  REQUIRE(trace(rrt) == Approx(1.0).epsilon(1e-7));

  // All those edge constraints...
  for (size_t i = 0; i < edges.n_cols; ++i)
  {
    REQUIRE(rrt(edges(0, i), edges(1, i)) == Approx(0.0).margin(1e-5));
    REQUIRE(rrt(edges(1, i), edges(0, i)) == Approx(0.0).margin(1e-5));
  }
}

/**
 * johnson8-4-4.co test case for Lovasz-Theta LRSDP.
 * See Monteiro and Burer 2004.  Uses arma::fmat.
 */
TEST_CASE("Johnson844LovaszThetaFMatSDP", "[LRSDPTest]")
{
  // Load the edges.
  arma::fmat edges;

  if (edges.load("data/johnson8-4-4.csv", arma::csv_ascii) == false)
  {
    FAIL("couldn't load data");
    return;
  }

  edges = edges.t();

  // The LRSDP itself and the initial point.
  arma::fmat coordinates;

  CreateLovaszThetaInitialPoint(edges, coordinates);

  LRSDP<SDP<arma::fmat>> lovasz(edges.n_cols + 1, 0, coordinates);

  SetupLovaszTheta(edges, lovasz);

  float finalValue = lovasz.Optimize(coordinates);

  // Final value taken from Monteiro + Burer 2004.
  REQUIRE(finalValue == Approx(-14.0).epsilon(0.1));

  // Now ensure that all the constraints are satisfied.
  arma::fmat rrt = coordinates * trans(coordinates);
  REQUIRE(trace(rrt) == Approx(1.0).epsilon(0.1));

  // All those edge constraints...
  for (size_t i = 0; i < edges.n_cols; ++i)
  {
    REQUIRE(rrt(edges(0, i), edges(1, i)) == Approx(0.0).margin(0.1));
    REQUIRE(rrt(edges(1, i), edges(0, i)) == Approx(0.0).margin(0.1));
  }
}


/**
 * Create an unweighted graph laplacian from the edges.
 */
void CreateSparseGraphLaplacian(const arma::mat& edges,
                                arma::sp_mat& laplacian)
{
  // Get the number of vertices in the problem.
  const size_t vertices = max(max(edges)) + 1;

  laplacian.zeros(vertices, vertices);

  for (size_t i = 0; i < edges.n_cols; ++i)
  {
    laplacian(edges(0, i), edges(1, i)) = -1.0;
    laplacian(edges(1, i), edges(0, i)) = -1.0;
  }

  for (size_t i = 0; i < vertices; ++i)
  {
    laplacian(i, i) = -arma::accu(laplacian.row(i));
  }
}

TEST_CASE("ErdosRenyiRandomGraphMaxCutSDP", "[LRSDPTest]")
{
  // Load the edges.
  arma::mat edges;

  if (edges.load("data/erdosrenyi-n100.csv", arma::csv_ascii) == false)
  {
    FAIL("couldn't load data");
    return;
  }

  edges = edges.t();

  arma::sp_mat laplacian;
  CreateSparseGraphLaplacian(edges, laplacian);

  float r = 0.5 + sqrt(0.25 + 2 * edges.n_cols);
  if (ceil(r) > laplacian.n_rows)
    r = laplacian.n_rows;

  // initialize coordinates to a feasible point
  arma::mat coordinates(laplacian.n_rows, ceil(r));
  coordinates.zeros();
  for (size_t i = 0; i < coordinates.n_rows; ++i)
  {
    coordinates(i, i % coordinates.n_cols) = 1.;
  }

  LRSDP<SDP<arma::sp_mat>> maxcut(laplacian.n_rows, 0, coordinates);
  maxcut.SDP().C() = laplacian;
  maxcut.SDP().C() *= -1.; // need to minimize the negative
  maxcut.SDP().SparseB().ones(laplacian.n_rows);
  for (size_t i = 0; i < laplacian.n_rows; ++i)
  {
    maxcut.SDP().SparseA()[i].zeros(laplacian.n_rows, laplacian.n_rows);
    maxcut.SDP().SparseA()[i](i, i) = 1.;
  }

  const double finalValue = maxcut.Optimize(coordinates);
  const arma::mat rrt = coordinates * trans(coordinates);

  for (size_t i = 0; i < laplacian.n_rows; ++i)
  {
    REQUIRE(rrt(i, i) == Approx(1.0).epsilon(1e-7));
  }

  // Final value taken by solving with Mosek
  REQUIRE(finalValue == Approx(-3672.7).epsilon(1e-3));
}

/*
 * Test a nuclear norm minimization SDP.
 *
 * Specifically, fix an unknown m x n matrix X. Our goal is to recover X from p
 * measurements of X, where the i-th measurement is of the form
 *
 *    b_i = dot(A_i, X)
 *
 * where the A_i's have iid entries from Normal(0, 1/p). We do this by solving
 * the the following semi-definite program
 *
 *    min ||X||_* subj to dot(A_i, X) = b_i, i=1,...,p
 *
 * where ||X||_* denotes the nuclear norm (sum of singular values) of X. The
 * equivalent SDP is
 *
 *    min tr(W1) + tr(W2) : [ W1, X ; X', W2 ] is PSD,
 *                          dot(A_i, X) = b_i, i = 1, ..., p
 *
 * For more details on matrix sensing and nuclear norm minimization, see
 *
 *    Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear
 *    Norm Minimization.
 *    Benjamin Recht, Maryam Fazel, Pablo Parrilo.
 *    SIAM Review 2010.
 *
 */
TEST_CASE("GaussianMatrixSensingSDP", "[LRSDPTest]")
{
  arma::mat Xorig, A;

  // read the unknown matrix X and the measurement matrices A_i in

  if (Xorig.load("data/sensing_X.csv", arma::csv_ascii) == false)
  {
    FAIL("couldn't load data");
  }

  if( A.load("data/sensing_A.csv", arma::csv_ascii) == false)
  {
    FAIL("couldn't load data");
  }

  const size_t m = Xorig.n_rows;
  const size_t n = Xorig.n_cols;
  const size_t p = A.n_rows;
  assert(A.n_cols == m * m);

  arma::vec b(p);
  for (size_t i = 0; i < p; ++i)
  {
    const arma::mat Ai = arma::reshape(A.row(i), n, m);
    b(i) = arma::dot(trans(Ai), Xorig);
  }

  float r = 0.5 + sqrt(0.25 + 2 * p);
  if (ceil(r) > m + n)
    r = m + n;

  arma::mat coordinates;
  coordinates.eye(m + n, ceil(r));

  LRSDP<SDP<arma::sp_mat>> sensing(0, p, coordinates, 15);
  sensing.SDP().C().eye(m + n, m + n);
  sensing.SDP().DenseB() = 2. * b;

  const auto blockRows = arma::span(0, m - 1);
  const auto blockCols = arma::span(m, m + n - 1);

  for (size_t i = 0; i < p; ++i)
  {
    const arma::mat Ai = arma::reshape(A.row(i), n, m);
    sensing.SDP().DenseA()[i].zeros(m + n, m + n);
    sensing.SDP().DenseA()[i](blockRows, blockCols) = trans(Ai);
    sensing.SDP().DenseA()[i](blockCols, blockRows) = Ai;
  }

  double finalValue = sensing.Optimize(coordinates);
  REQUIRE(finalValue == Approx(44.7550132629).epsilon(1e-3));

  const arma::mat rrt = coordinates * trans(coordinates);
  for (size_t i = 0; i < p; ++i)
  {
    const arma::mat Ai = arma::reshape(A.row(i), n, m);
    const double measurement =
        arma::dot(trans(Ai), rrt(blockRows, blockCols));
    REQUIRE(measurement == Approx(b(i)).epsilon(0.0006));
  }

  // check matrix recovery
  const double err = arma::norm(Xorig - rrt(blockRows, blockCols), "fro") /
      arma::norm(Xorig, "fro");
  REQUIRE(err == Approx(0.0).margin(0.05));
}