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annotate liboctave/oct-norm.cc @ 10145:fa01c1670b3e

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make p-norms breakable
author Jaroslav Hajek <highegg@gmail.com>
date Wed, 20 Jan 2010 12:24:00 +0100
parents 829e69ec3110
children 12884915a8e4
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1 /*
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2
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16f53d29049f update copyright notices
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3 Copyright (C) 2008, 2009 VZLU Prague, a.s.
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4
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5 This file is part of Octave.
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6
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7 Octave is free software; you can redistribute it and/or modify it
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8 under the terms of the GNU General Public License as published by the
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9 Free Software Foundation; either version 3 of the License, or (at your
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10 option) any later version.
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11
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12 Octave is distributed in the hope that it will be useful, but WITHOUT
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13 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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14 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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15 for more details.
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16
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17 You should have received a copy of the GNU General Public License
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18 along with Octave; see the file COPYING. If not, see
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19 <http://www.gnu.org/licenses/>.
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20
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21 */
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22
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23 // author: Jaroslav Hajek <highegg@gmail.com>
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24
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25 #ifdef HAVE_CONFIG_H
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26 #include <config.h>
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27 #endif
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28
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29 #include <cassert>
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30 #include <cfloat>
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31 #include <cmath>
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32
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33 #include <iostream>
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34 #include <vector>
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35
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36 #include "oct-cmplx.h"
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37 #include "lo-error.h"
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38 #include "lo-ieee.h"
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39 #include "Array.h"
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40 #include "Array-util.h"
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41 #include "CMatrix.h"
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42 #include "dMatrix.h"
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43 #include "fCMatrix.h"
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44 #include "fMatrix.h"
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45 #include "CColVector.h"
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46 #include "dColVector.h"
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47 #include "CRowVector.h"
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48 #include "dRowVector.h"
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49 #include "fCColVector.h"
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50 #include "fColVector.h"
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51 #include "fCRowVector.h"
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52 #include "fRowVector.h"
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53 #include "CSparse.h"
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54 #include "dSparse.h"
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55 #include "dbleSVD.h"
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56 #include "CmplxSVD.h"
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57 #include "floatSVD.h"
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58 #include "fCmplxSVD.h"
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59
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60 // Theory: norm accumulator is an object that has an accum method able to handle
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61 // both real and complex element, and a cast operator returning the intermediate
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62 // norm.
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63
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64 // norm accumulator for the p-norm
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65 template <class R>
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66 class norm_accumulator_p
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67 {
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68 R p,scl,sum;
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69 public:
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70 norm_accumulator_p () {} // we need this one for Array
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71 norm_accumulator_p (R pp) : p(pp), scl(0), sum(1) {}
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72
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73 template<class U>
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74 void accum (U val)
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75 {
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76 octave_quit ();
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77 R t = std::abs (val);
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78 if (scl == t) // we need this to handle Infs properly
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79 sum += 1;
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80 else if (scl < t)
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81 {
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82 sum *= std::pow (scl/t, p);
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83 sum += 1;
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84 scl = t;
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85 }
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86 else if (t != 0)
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87 sum += std::pow (t/scl, p);
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88 }
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89 operator R () { return scl * std::pow (sum, 1/p); }
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90 };
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91
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92 // norm accumulator for the minus p-pseudonorm
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93 template <class R>
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94 class norm_accumulator_mp
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95 {
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96 R p,scl,sum;
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97 public:
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98 norm_accumulator_mp () {} // we need this one for Array
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99 norm_accumulator_mp (R pp) : p(pp), scl(0), sum(1) {}
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100
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101 template<class U>
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102 void accum (U val)
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103 {
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104 octave_quit ();
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105 R t = 1 / std::abs (val);
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106 if (scl == t)
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107 sum += 1;
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108 else if (scl < t)
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109 {
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110 sum *= std::pow (scl/t, p);
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111 sum += 1;
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112 scl = t;
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113 }
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114 else if (t != 0)
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115 sum += std::pow (t/scl, p);
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116 }
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117 operator R () { return scl * std::pow (sum, -1/p); }
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118 };
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119
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120 // norm accumulator for the 2-norm (euclidean)
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121 template <class R>
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122 class norm_accumulator_2
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123 {
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124 R scl,sum;
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125 static R pow2 (R x) { return x*x; }
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126 public:
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127 norm_accumulator_2 () : scl(0), sum(1) {}
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128
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129 void accum (R val)
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130 {
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131 R t = std::abs (val);
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132 if (scl == t)
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133 sum += 1;
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134 else if (scl < t)
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135 {
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136 sum *= pow2 (scl/t);
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137 sum += 1;
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138 scl = t;
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139 }
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140 else if (t != 0)
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141 sum += pow2 (t/scl);
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142 }
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143
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144 void accum (std::complex<R> val)
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145 {
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146 accum (val.real ());
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147 accum (val.imag ());
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148 }
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149
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150 operator R () { return scl * std::sqrt (sum); }
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151 };
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152
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153 // norm accumulator for the 1-norm (city metric)
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154 template <class R>
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155 class norm_accumulator_1
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156 {
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157 R sum;
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158 public:
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159 norm_accumulator_1 () : sum (0) {}
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160 template<class U>
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161 void accum (U val)
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162 {
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163 sum += std::abs (val);
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164 }
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165 operator R () { return sum; }
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166 };
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167
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168 // norm accumulator for the inf-norm (max metric)
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169 template <class R>
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170 class norm_accumulator_inf
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171 {
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172 R max;
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173 public:
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174 norm_accumulator_inf () : max (0) {}
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175 template<class U>
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176 void accum (U val)
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177 {
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178 max = std::max (max, std::abs (val));
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179 }
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180 operator R () { return max; }
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181 };
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182
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183 // norm accumulator for the -inf pseudonorm (min abs value)
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184 template <class R>
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185 class norm_accumulator_minf
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186 {
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187 R min;
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188 public:
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189 norm_accumulator_minf () : min (octave_Inf) {}
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190 template<class U>
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191 void accum (U val)
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192 {
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193 min = std::min (min, std::abs (val));
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194 }
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195 operator R () { return min; }
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196 };
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197
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198 // norm accumulator for the 0-pseudonorm (hamming distance)
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199 template <class R>
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200 class norm_accumulator_0
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201 {
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202 unsigned int num;
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203 public:
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204 norm_accumulator_0 () : num (0) {}
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205 template<class U>
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206 void accum (U val)
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207 {
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208 if (val != static_cast<U> (0)) ++num;
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209 }
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210 operator R () { return num; }
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211 };
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212
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213
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214 // OK, we're armed :) Now let's go for the fun
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215
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216 template <class T, class R, class ACC>
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217 inline void vector_norm (const Array<T>& v, R& res, ACC acc)
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218 {
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219 for (octave_idx_type i = 0; i < v.numel (); i++)
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220 acc.accum (v(i));
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221
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222 res = acc;
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223 }
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224
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225 // dense versions
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226 template <class T, class R, class ACC>
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227 void column_norms (const MArray2<T>& m, MArray<R>& res, ACC acc)
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228 {
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229 res = MArray2<R> (1, m.columns ());
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230 for (octave_idx_type j = 0; j < m.columns (); j++)
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231 {
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232 ACC accj = acc;
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233 for (octave_idx_type i = 0; i < m.rows (); i++)
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234 accj.accum (m(i, j));
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235
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236 res.xelem (j) = accj;
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237 }
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238 }
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239
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240 template <class T, class R, class ACC>
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241 void row_norms (const MArray2<T>& m, MArray<R>& res, ACC acc)
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242 {
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243 res = MArray2<R> (m.rows (), 1);
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244 std::vector<ACC> acci (m.rows (), acc);
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245 for (octave_idx_type j = 0; j < m.columns (); j++)
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246 {
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247 for (octave_idx_type i = 0; i < m.rows (); i++)
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248 acci[i].accum (m(i, j));
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249 }
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250
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251 for (octave_idx_type i = 0; i < m.rows (); i++)
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252 res.xelem (i) = acci[i];
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253 }
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254
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255 // sparse versions
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256 template <class T, class R, class ACC>
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257 void column_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
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258 {
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259 res = MArray2<R> (1, m.columns ());
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260 for (octave_idx_type j = 0; j < m.columns (); j++)
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261 {
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262 ACC accj = acc;
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263 for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
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264 accj.accum (m.data (k));
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265
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266 res.xelem (j) = accj;
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267 }
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268 }
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269
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270 template <class T, class R, class ACC>
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271 void row_norms (const MSparse<T>& m, MArray<R>& res, ACC acc)
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272 {
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273 res = MArray2<R> (m.rows (), 1);
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274 std::vector<ACC> acci (m.rows (), acc);
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275 for (octave_idx_type j = 0; j < m.columns (); j++)
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276 {
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277 for (octave_idx_type k = m.cidx (j); k < m.cidx (j+1); k++)
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278 acci[m.ridx (k)].accum (m.data (k));
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279 }
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280
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281 for (octave_idx_type i = 0; i < m.rows (); i++)
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282 res.xelem (i) = acci[i];
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283 }
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284
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285 // now the dispatchers
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286 #define DEFINE_DISPATCHER(FUNC_NAME, ARG_TYPE, RES_TYPE) \
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287 template <class T, class R> \
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288 RES_TYPE FUNC_NAME (const ARG_TYPE& v, R p) \
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289 { \
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290 RES_TYPE res; \
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291 if (p == 2) \
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292 FUNC_NAME (v, res, norm_accumulator_2<R> ()); \
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293 else if (p == 1) \
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294 FUNC_NAME (v, res, norm_accumulator_1<R> ()); \
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295 else if (lo_ieee_isinf (p)) \
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296 { \
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297 if (p > 0) \
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298 FUNC_NAME (v, res, norm_accumulator_inf<R> ()); \
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299 else \
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300 FUNC_NAME (v, res, norm_accumulator_minf<R> ()); \
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301 } \
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302 else if (p == 0) \
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303 FUNC_NAME (v, res, norm_accumulator_0<R> ()); \
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304 else if (p > 0) \
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305 FUNC_NAME (v, res, norm_accumulator_p<R> (p)); \
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306 else \
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307 FUNC_NAME (v, res, norm_accumulator_mp<R> (p)); \
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308 return res; \
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309 }
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310
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311 DEFINE_DISPATCHER (vector_norm, MArray<T>, R)
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312 DEFINE_DISPATCHER (vector_norm, MArray2<T>, R)
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313 DEFINE_DISPATCHER (column_norms, MArray2<T>, MArray<R>)
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314 DEFINE_DISPATCHER (row_norms, MArray2<T>, MArray<R>)
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315 DEFINE_DISPATCHER (column_norms, MSparse<T>, MArray<R>)
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316 DEFINE_DISPATCHER (row_norms, MSparse<T>, MArray<R>)
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317
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318 // The approximate subproblem in Higham's method. Find lambda and mu such that
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319 // norm ([lambda, mu], p) == 1 and norm (y*lambda + col*mu, p) is maximized.
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320 // Real version. As in Higham's paper.
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321 template <class ColVectorT, class R>
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322 static void
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323 higham_subp (const ColVectorT& y, const ColVectorT& col,
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324 octave_idx_type nsamp, R p, R& lambda, R& mu)
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325 {
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326 R nrm = 0;
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327 for (octave_idx_type i = 0; i < nsamp; i++)
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328 {
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329 octave_quit ();
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330 R fi = i*M_PI/nsamp, lambda1 = cos (fi), mu1 = sin (fi);
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331 R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
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332 std::pow (std::abs (mu1), p), 1/p);
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333 lambda1 /= lmnr; mu1 /= lmnr;
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334 R nrm1 = vector_norm (lambda1 * y + mu1 * col, p);
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335 if (nrm1 > nrm)
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336 {
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337 lambda = lambda1;
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338 mu = mu1;
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339 nrm = nrm1;
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340 }
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341 }
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342 }
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343
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344 // Complex version. Higham's paper does not deal with complex case, so we use a simple
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345 // extension. First, guess the magnitudes as in real version, then try to rotate lambda
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346 // to improve further.
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347 template <class ColVectorT, class R>
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348 static void
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349 higham_subp (const ColVectorT& y, const ColVectorT& col,
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350 octave_idx_type nsamp, R p,
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351 std::complex<R>& lambda, std::complex<R>& mu)
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352 {
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353 typedef std::complex<R> CR;
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354 R nrm = 0;
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355 lambda = 1.0;
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356 CR lamcu = lambda / std::abs (lambda);
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357 // Probe magnitudes
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358 for (octave_idx_type i = 0; i < nsamp; i++)
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359 {
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360 octave_quit ();
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361 R fi = i*M_PI/nsamp, lambda1 = cos (fi), mu1 = sin (fi);
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362 R lmnr = std::pow (std::pow (std::abs (lambda1), p) +
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363 std::pow (std::abs (mu1), p), 1/p);
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364 lambda1 /= lmnr; mu1 /= lmnr;
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365 R nrm1 = vector_norm (lambda1 * lamcu * y + mu1 * col, p);
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366 if (nrm1 > nrm)
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367 {
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368 lambda = lambda1 * lamcu;
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369 mu = mu1;
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370 nrm = nrm1;
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371 }
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372 }
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373 R lama = std::abs (lambda);
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374 // Probe orientation
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375 for (octave_idx_type i = 0; i < nsamp; i++)
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376 {
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377 octave_quit ();
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378 R fi = i*M_PI/nsamp;
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379 lamcu = CR (cos (fi), sin (fi));
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380 R nrm1 = vector_norm (lama * lamcu * y + mu * col, p);
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381 if (nrm1 > nrm)
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382 {
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383 lambda = lama * lamcu;
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384 nrm = nrm1;
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385 }
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386 }
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387 }
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388
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389 // the p-dual element (should work for both real and complex)
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390 template <class T, class R>
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391 inline T elem_dual_p (T x, R p)
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392 {
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393 return signum (x) * std::pow (std::abs (x), p-1);
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394 }
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395
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396 // the VectorT is used for vectors, but actually it has to be
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397 // a Matrix type to allow all the operations. For instance SparseMatrix
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398 // does not support multiplication with column/row vectors.
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399 // the dual vector
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400 template <class VectorT, class R>
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401 VectorT dual_p (const VectorT& x, R p, R q)
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402 {
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403 VectorT res (x.dims ());
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404 for (octave_idx_type i = 0; i < x.numel (); i++)
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405 res.xelem (i) = elem_dual_p (x(i), p);
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406 return res / vector_norm (res, q);
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407 }
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408
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409 // Higham's hybrid method
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410 template <class MatrixT, class VectorT, class R>
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411 R higham (const MatrixT& m, R p, R tol, int maxiter,
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412 VectorT& x)
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413 {
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414 x.resize (m.columns (), 1);
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415 // the OSE part
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416 VectorT y(m.rows (), 1, 0), z(m.rows (), 1);
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417 typedef typename VectorT::element_type RR;
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418 RR lambda = 0, mu = 0;
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419 for (octave_idx_type k = 0; k < m.columns (); k++)
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420 {
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421 octave_quit ();
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422 VectorT col (m.column (k));
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423 if (k > 0)
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424 higham_subp (y, col, 4*k, p, lambda, mu);
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425 for (octave_idx_type i = 0; i < k; i++)
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426 x(i) *= lambda;
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427 x(k) = mu;
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428 y = lambda * y + mu * col;
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429 }
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430
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431 // the PM part
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432 x = x / vector_norm (x, p);
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433 R q = p/(p-1);
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434
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435 R gamma = 0, gamma1;
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436 int iter = 0;
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437 while (iter < maxiter)
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438 {
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439 octave_quit ();
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440 y = m*x;
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441 gamma1 = gamma;
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442 gamma = vector_norm (y, p);
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443 z = dual_p (y, p, q);
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444 z = z.hermitian ();
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445 z = z * m;
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446
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447 if (iter > 0 && (vector_norm (z, q) <= gamma
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448 || (gamma - gamma1) <= tol*gamma))
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449 break;
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450
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451 z = z.hermitian ();
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452 x = dual_p (z, q, p);
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453 iter ++;
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454 }
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455
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456 return gamma;
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457 }
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458
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459 // derive column vector and SVD types
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460
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461 static const char *p_less1_gripe = "xnorm: p must be at least 1";
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462
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463 // Static constant to control the maximum number of iterations. 100 seems to be a good value.
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464 // Eventually, we can provide a means to change this constant from Octave.
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465 static int max_norm_iter = 100;
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466
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467 // version with SVD for dense matrices
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468 template <class MatrixT, class VectorT, class SVDT, class R>
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469 R matrix_norm (const MatrixT& m, R p, VectorT, SVDT)
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470 {
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471 R res = 0;
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472 if (p == 2)
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473 {
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474 SVDT svd (m, SVD::sigma_only);
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475 res = svd.singular_values () (0,0);
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476 }
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477 else if (p == 1)
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478 res = xcolnorms (m, 1).max ();
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479 else if (lo_ieee_isinf (p))
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480 res = xrownorms (m, 1).max ();
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481 else if (p > 1)
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482 {
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483 VectorT x;
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484 const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
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485 res = higham (m, p, sqrteps, max_norm_iter, x);
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486 }
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487 else
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488 (*current_liboctave_error_handler) (p_less1_gripe);
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489
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490 return res;
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491 }
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492
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493 // SVD-free version for sparse matrices
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494 template <class MatrixT, class VectorT, class R>
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495 R matrix_norm (const MatrixT& m, R p, VectorT)
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496 {
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497 R res = 0;
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498 if (p == 1)
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499 res = xcolnorms (m, 1).max ();
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500 else if (lo_ieee_isinf (p))
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501 res = xrownorms (m, 1).max ();
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502 else if (p > 1)
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503 {
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504 VectorT x;
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505 const R sqrteps = std::sqrt (std::numeric_limits<R>::epsilon ());
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506 res = higham (m, p, sqrteps, max_norm_iter, x);
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507 }
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508 else
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509 (*current_liboctave_error_handler) (p_less1_gripe);
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510
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511 return res;
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512 }
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513
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514 // and finally, here's what we've promised in the header file
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515
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516 #define DEFINE_XNORM_FUNCS(PREFIX, RTYPE) \
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517 OCTAVE_API RTYPE xnorm (const PREFIX##ColumnVector& x, RTYPE p) \
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518 { return vector_norm (x, p); } \
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519 OCTAVE_API RTYPE xnorm (const PREFIX##RowVector& x, RTYPE p) \
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520 { return vector_norm (x, p); } \
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521 OCTAVE_API RTYPE xnorm (const PREFIX##Matrix& x, RTYPE p) \
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522 { return matrix_norm (x, p, PREFIX##Matrix (), PREFIX##SVD ()); } \
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523 OCTAVE_API RTYPE xfrobnorm (const PREFIX##Matrix& x) \
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524 { return vector_norm (x, static_cast<RTYPE> (2)); }
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525
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526 DEFINE_XNORM_FUNCS(, double)
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527 DEFINE_XNORM_FUNCS(Complex, double)
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528 DEFINE_XNORM_FUNCS(Float, float)
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529 DEFINE_XNORM_FUNCS(FloatComplex, float)
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530
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531 // this is needed to avoid copying the sparse matrix for xfrobnorm
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532 template <class T, class R>
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533 inline void array_norm_2 (const T* v, octave_idx_type n, R& res)
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534 {
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535 norm_accumulator_2<R> acc;
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536 for (octave_idx_type i = 0; i < n; i++)
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537 acc.accum (v[i]);
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538
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539 res = acc;
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540 }
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541
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542 #define DEFINE_XNORM_SPARSE_FUNCS(PREFIX, RTYPE) \
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543 OCTAVE_API RTYPE xnorm (const Sparse##PREFIX##Matrix& x, RTYPE p) \
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544 { return matrix_norm (x, p, PREFIX##Matrix ()); } \
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545 OCTAVE_API RTYPE xfrobnorm (const Sparse##PREFIX##Matrix& x) \
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546 { \
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547 RTYPE res; \
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548 array_norm_2 (x.data (), x.nnz (), res); \
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549 return res; \
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550 }
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551
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552 DEFINE_XNORM_SPARSE_FUNCS(, double)
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553 DEFINE_XNORM_SPARSE_FUNCS(Complex, double)
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554
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555 #define DEFINE_COLROW_NORM_FUNCS(PREFIX, RPREFIX, RTYPE) \
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556 extern OCTAVE_API RPREFIX##RowVector xcolnorms (const PREFIX##Matrix& m, RTYPE p) \
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557 { return column_norms (m, p); } \
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558 extern OCTAVE_API RPREFIX##ColumnVector xrownorms (const PREFIX##Matrix& m, RTYPE p) \
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559 { return row_norms (m, p); } \
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560
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561 DEFINE_COLROW_NORM_FUNCS(, , double)
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562 DEFINE_COLROW_NORM_FUNCS(Complex, , double)
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563 DEFINE_COLROW_NORM_FUNCS(Float, Float, float)
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564 DEFINE_COLROW_NORM_FUNCS(FloatComplex, Float, float)
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565
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566 DEFINE_COLROW_NORM_FUNCS(Sparse, , double)
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567 DEFINE_COLROW_NORM_FUNCS(SparseComplex, , double)
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568