1 // Ceres Solver - A fast non-linear least squares minimizer
2 // Copyright 2012 Google Inc. All rights reserved.
3 // http://code.google.com/p/ceres-solver/
4 //
5 // Redistribution and use in source and binary forms, with or without
6 // modification, are permitted provided that the following conditions are met:
7 //
8 // * Redistributions of source code must retain the above copyright notice,
9 // this list of conditions and the following disclaimer.
10 // * Redistributions in binary form must reproduce the above copyright notice,
11 // this list of conditions and the following disclaimer in the documentation
12 // and/or other materials provided with the distribution.
13 // * Neither the name of Google Inc. nor the names of its contributors may be
14 // used to endorse or promote products derived from this software without
15 // specific prior written permission.
16 //
17 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
18 // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
19 // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
20 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
21 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
22 // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
23 // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
24 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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26 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
27 // POSSIBILITY OF SUCH DAMAGE.
28 //
29 // Author: moll.markus@arcor.de (Markus Moll)
30
31 #include <limits>
32 #include "ceres/internal/eigen.h"
33 #include "ceres/internal/scoped_ptr.h"
34 #include "ceres/dense_qr_solver.h"
35 #include "ceres/dogleg_strategy.h"
36 #include "ceres/linear_solver.h"
37 #include "ceres/trust_region_strategy.h"
38 #include "glog/logging.h"
39 #include "gtest/gtest.h"
40
41 namespace ceres {
42 namespace internal {
43 namespace {
44
45 class Fixture : public testing::Test {
46 protected:
47 scoped_ptr<DenseSparseMatrix> jacobian_;
48 Vector residual_;
49 Vector x_;
50 TrustRegionStrategy::Options options_;
51 };
52
53 // A test problem where
54 //
55 // J^T J = Q diag([1 2 4 8 16 32]) Q^T
56 //
57 // where Q is a randomly chosen orthonormal basis of R^6.
58 // The residual is chosen so that the minimum of the quadratic function is
59 // at (1, 1, 1, 1, 1, 1). It is therefore at a distance of sqrt(6) ~ 2.45
60 // from the origin.
61 class DoglegStrategyFixtureEllipse : public Fixture {
62 protected:
SetUp()63 virtual void SetUp() {
64 Matrix basis(6, 6);
65 // The following lines exceed 80 characters for better readability.
66 basis << -0.1046920933796121, -0.7449367449921986, -0.4190744502875876, -0.4480450716142566, 0.2375351607929440, -0.0363053418882862,
67 0.4064975684355914, 0.2681113508511354, -0.7463625494601520, -0.0803264850508117, -0.4463149623021321, 0.0130224954867195,
68 -0.5514387729089798, 0.1026621026168657, -0.5008316122125011, 0.5738122212666414, 0.2974664724007106, 0.1296020877535158,
69 0.5037835370947156, 0.2668479925183712, -0.1051754618492798, -0.0272739396578799, 0.7947481647088278, -0.1776623363955670,
70 -0.4005458426625444, 0.2939330589634109, -0.0682629380550051, -0.2895448882503687, -0.0457239396341685, -0.8139899477847840,
71 -0.3247764582762654, 0.4528151365941945, -0.0276683863102816, -0.6155994592510784, 0.1489240599972848, 0.5362574892189350;
72
73 Vector Ddiag(6);
74 Ddiag << 1.0, 2.0, 4.0, 8.0, 16.0, 32.0;
75
76 Matrix sqrtD = Ddiag.array().sqrt().matrix().asDiagonal();
77 Matrix jacobian = sqrtD * basis;
78 jacobian_.reset(new DenseSparseMatrix(jacobian));
79
80 Vector minimum(6);
81 minimum << 1.0, 1.0, 1.0, 1.0, 1.0, 1.0;
82 residual_ = -jacobian * minimum;
83
84 x_.resize(6);
85 x_.setZero();
86
87 options_.min_lm_diagonal = 1.0;
88 options_.max_lm_diagonal = 1.0;
89 }
90 };
91
92 // A test problem where
93 //
94 // J^T J = diag([1 2 4 8 16 32]) .
95 //
96 // The residual is chosen so that the minimum of the quadratic function is
97 // at (0, 0, 1, 0, 0, 0). It is therefore at a distance of 1 from the origin.
98 // The gradient at the origin points towards the global minimum.
99 class DoglegStrategyFixtureValley : public Fixture {
100 protected:
SetUp()101 virtual void SetUp() {
102 Vector Ddiag(6);
103 Ddiag << 1.0, 2.0, 4.0, 8.0, 16.0, 32.0;
104
105 Matrix jacobian = Ddiag.asDiagonal();
106 jacobian_.reset(new DenseSparseMatrix(jacobian));
107
108 Vector minimum(6);
109 minimum << 0.0, 0.0, 1.0, 0.0, 0.0, 0.0;
110 residual_ = -jacobian * minimum;
111
112 x_.resize(6);
113 x_.setZero();
114
115 options_.min_lm_diagonal = 1.0;
116 options_.max_lm_diagonal = 1.0;
117 }
118 };
119
120 const double kTolerance = 1e-14;
121 const double kToleranceLoose = 1e-5;
122 const double kEpsilon = std::numeric_limits<double>::epsilon();
123
124 } // namespace
125
126 // The DoglegStrategy must never return a step that is longer than the current
127 // trust region radius.
TEST_F(DoglegStrategyFixtureEllipse,TrustRegionObeyedTraditional)128 TEST_F(DoglegStrategyFixtureEllipse, TrustRegionObeyedTraditional) {
129 scoped_ptr<LinearSolver> linear_solver(
130 new DenseQRSolver(LinearSolver::Options()));
131 options_.linear_solver = linear_solver.get();
132 // The global minimum is at (1, 1, ..., 1), so the distance to it is
133 // sqrt(6.0). By restricting the trust region to a radius of 2.0,
134 // we test if the trust region is actually obeyed.
135 options_.dogleg_type = TRADITIONAL_DOGLEG;
136 options_.initial_radius = 2.0;
137 options_.max_radius = 2.0;
138
139 DoglegStrategy strategy(options_);
140 TrustRegionStrategy::PerSolveOptions pso;
141
142 TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
143 jacobian_.get(),
144 residual_.data(),
145 x_.data());
146
147 EXPECT_NE(summary.termination_type, LINEAR_SOLVER_FAILURE);
148 EXPECT_LE(x_.norm(), options_.initial_radius * (1.0 + 4.0 * kEpsilon));
149 }
150
TEST_F(DoglegStrategyFixtureEllipse,TrustRegionObeyedSubspace)151 TEST_F(DoglegStrategyFixtureEllipse, TrustRegionObeyedSubspace) {
152 scoped_ptr<LinearSolver> linear_solver(
153 new DenseQRSolver(LinearSolver::Options()));
154 options_.linear_solver = linear_solver.get();
155 options_.dogleg_type = SUBSPACE_DOGLEG;
156 options_.initial_radius = 2.0;
157 options_.max_radius = 2.0;
158
159 DoglegStrategy strategy(options_);
160 TrustRegionStrategy::PerSolveOptions pso;
161
162 TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
163 jacobian_.get(),
164 residual_.data(),
165 x_.data());
166
167 EXPECT_NE(summary.termination_type, LINEAR_SOLVER_FAILURE);
168 EXPECT_LE(x_.norm(), options_.initial_radius * (1.0 + 4.0 * kEpsilon));
169 }
170
TEST_F(DoglegStrategyFixtureEllipse,CorrectGaussNewtonStep)171 TEST_F(DoglegStrategyFixtureEllipse, CorrectGaussNewtonStep) {
172 scoped_ptr<LinearSolver> linear_solver(
173 new DenseQRSolver(LinearSolver::Options()));
174 options_.linear_solver = linear_solver.get();
175 options_.dogleg_type = SUBSPACE_DOGLEG;
176 options_.initial_radius = 10.0;
177 options_.max_radius = 10.0;
178
179 DoglegStrategy strategy(options_);
180 TrustRegionStrategy::PerSolveOptions pso;
181
182 TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
183 jacobian_.get(),
184 residual_.data(),
185 x_.data());
186
187 EXPECT_NE(summary.termination_type, LINEAR_SOLVER_FAILURE);
188 EXPECT_NEAR(x_(0), 1.0, kToleranceLoose);
189 EXPECT_NEAR(x_(1), 1.0, kToleranceLoose);
190 EXPECT_NEAR(x_(2), 1.0, kToleranceLoose);
191 EXPECT_NEAR(x_(3), 1.0, kToleranceLoose);
192 EXPECT_NEAR(x_(4), 1.0, kToleranceLoose);
193 EXPECT_NEAR(x_(5), 1.0, kToleranceLoose);
194 }
195
196 // Test if the subspace basis is a valid orthonormal basis of the space spanned
197 // by the gradient and the Gauss-Newton point.
TEST_F(DoglegStrategyFixtureEllipse,ValidSubspaceBasis)198 TEST_F(DoglegStrategyFixtureEllipse, ValidSubspaceBasis) {
199 scoped_ptr<LinearSolver> linear_solver(
200 new DenseQRSolver(LinearSolver::Options()));
201 options_.linear_solver = linear_solver.get();
202 options_.dogleg_type = SUBSPACE_DOGLEG;
203 options_.initial_radius = 2.0;
204 options_.max_radius = 2.0;
205
206 DoglegStrategy strategy(options_);
207 TrustRegionStrategy::PerSolveOptions pso;
208
209 strategy.ComputeStep(pso, jacobian_.get(), residual_.data(), x_.data());
210
211 // Check if the basis is orthonormal.
212 const Matrix basis = strategy.subspace_basis();
213 EXPECT_NEAR(basis.col(0).norm(), 1.0, kTolerance);
214 EXPECT_NEAR(basis.col(1).norm(), 1.0, kTolerance);
215 EXPECT_NEAR(basis.col(0).dot(basis.col(1)), 0.0, kTolerance);
216
217 // Check if the gradient projects onto itself.
218 const Vector gradient = strategy.gradient();
219 EXPECT_NEAR((gradient - basis*(basis.transpose()*gradient)).norm(),
220 0.0,
221 kTolerance);
222
223 // Check if the Gauss-Newton point projects onto itself.
224 const Vector gn = strategy.gauss_newton_step();
225 EXPECT_NEAR((gn - basis*(basis.transpose()*gn)).norm(),
226 0.0,
227 kTolerance);
228 }
229
230 // Test if the step is correct if the gradient and the Gauss-Newton step point
231 // in the same direction and the Gauss-Newton step is outside the trust region,
232 // i.e. the trust region is active.
TEST_F(DoglegStrategyFixtureValley,CorrectStepLocalOptimumAlongGradient)233 TEST_F(DoglegStrategyFixtureValley, CorrectStepLocalOptimumAlongGradient) {
234 scoped_ptr<LinearSolver> linear_solver(
235 new DenseQRSolver(LinearSolver::Options()));
236 options_.linear_solver = linear_solver.get();
237 options_.dogleg_type = SUBSPACE_DOGLEG;
238 options_.initial_radius = 0.25;
239 options_.max_radius = 0.25;
240
241 DoglegStrategy strategy(options_);
242 TrustRegionStrategy::PerSolveOptions pso;
243
244 TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
245 jacobian_.get(),
246 residual_.data(),
247 x_.data());
248
249 EXPECT_NE(summary.termination_type, LINEAR_SOLVER_FAILURE);
250 EXPECT_NEAR(x_(0), 0.0, kToleranceLoose);
251 EXPECT_NEAR(x_(1), 0.0, kToleranceLoose);
252 EXPECT_NEAR(x_(2), options_.initial_radius, kToleranceLoose);
253 EXPECT_NEAR(x_(3), 0.0, kToleranceLoose);
254 EXPECT_NEAR(x_(4), 0.0, kToleranceLoose);
255 EXPECT_NEAR(x_(5), 0.0, kToleranceLoose);
256 }
257
258 // Test if the step is correct if the gradient and the Gauss-Newton step point
259 // in the same direction and the Gauss-Newton step is inside the trust region,
260 // i.e. the trust region is inactive.
TEST_F(DoglegStrategyFixtureValley,CorrectStepGlobalOptimumAlongGradient)261 TEST_F(DoglegStrategyFixtureValley, CorrectStepGlobalOptimumAlongGradient) {
262 scoped_ptr<LinearSolver> linear_solver(
263 new DenseQRSolver(LinearSolver::Options()));
264 options_.linear_solver = linear_solver.get();
265 options_.dogleg_type = SUBSPACE_DOGLEG;
266 options_.initial_radius = 2.0;
267 options_.max_radius = 2.0;
268
269 DoglegStrategy strategy(options_);
270 TrustRegionStrategy::PerSolveOptions pso;
271
272 TrustRegionStrategy::Summary summary = strategy.ComputeStep(pso,
273 jacobian_.get(),
274 residual_.data(),
275 x_.data());
276
277 EXPECT_NE(summary.termination_type, LINEAR_SOLVER_FAILURE);
278 EXPECT_NEAR(x_(0), 0.0, kToleranceLoose);
279 EXPECT_NEAR(x_(1), 0.0, kToleranceLoose);
280 EXPECT_NEAR(x_(2), 1.0, kToleranceLoose);
281 EXPECT_NEAR(x_(3), 0.0, kToleranceLoose);
282 EXPECT_NEAR(x_(4), 0.0, kToleranceLoose);
283 EXPECT_NEAR(x_(5), 0.0, kToleranceLoose);
284 }
285
286 } // namespace internal
287 } // namespace ceres
288