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‎README.md‎

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‎pictures/linear_regression.gif‎

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‎pictures/logistic_regression.gif‎

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‎pictures/network-1.png‎

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‎pictures/network-2.png‎

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‎pictures/training_data_1.png‎

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‎pictures/training_data_2.png‎

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‎tst/example-in-readme-1.tst‎

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gap> LoadPackage( "MachineLearningForCAP" );
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true
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gap> Para := CategoryOfParametrisedMorphisms( SkeletalSmoothMaps );
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CategoryOfParametrisedMorphisms( SkeletalSmoothMaps )
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gap> training_set := [ [ 1, 2.9 ], [ 2, 5.1 ], [ 3, 7.05 ] ];
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[ [ 1, 2.9 ], [ 2, 5.1 ], [ 3, 7.05 ] ]
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gap> input_dim := 1;; output_dim := 1;; hidden_dims := [ ];;
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gap> f := PredictionMorphismOfNeuralNetwork( Para, input_dim, hidden_dims, output_dim, "IdFunc" );;
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gap> Display( f );
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ℝ^1 -> ℝ^1 defined by:
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Underlying Object:
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-----------------
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ℝ^2
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Underlying Morphism:
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-------------------
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ℝ^3 -> ℝ^1
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‣ x1 * x3 + x2
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gap> parameters := [ 2, 1 ];; x := [ 2 ];;
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gap> Eval( f, [ parameters, x ] );
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[ 5 ]
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gap> ell := LossMorphismOfNeuralNetwork( Para, input_dim, hidden_dims, output_dim, "IdFunc" );;
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gap> Display( ell );
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ℝ^2 -> ℝ^1 defined by:
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Underlying Object:
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-----------------
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ℝ^2
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Underlying Morphism:
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-------------------
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ℝ^4 -> ℝ^1
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‣ (x1 * x3 + x2 - x4) ^ 2 / 1
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gap> Lenses := CategoryOfLenses( SkeletalSmoothMaps );
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CategoryOfLenses( SkeletalSmoothMaps )
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gap> optimizer := Lenses.GradientDescentOptimizer( : learning_rate := 0.01 );;
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gap> batch_size := 1;;
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gap> one_epoch_update := OneEpochUpdateLens( ell, optimizer, training_set, batch_size );;
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gap> parameters := [ 0.1, -0.1 ];;
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gap> nr_epochs := 15;;
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gap> parameters := Fit( one_epoch_update, nr_epochs, parameters );
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Epoch 0/15 - loss = 26.777499999999993
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Epoch 1/15 - loss = 13.002145872163197
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Epoch 2/15 - loss = 6.3171942035316935
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Epoch 3/15 - loss = 3.0722513061917534
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Epoch 4/15 - loss = 1.4965356389126505
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Epoch 5/15 - loss = 0.73097379078374358
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Epoch 6/15 - loss = 0.35874171019291579
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Epoch 7/15 - loss = 0.1775574969062125
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Epoch 8/15 - loss = 0.089228700384937534
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Epoch 9/15 - loss = 0.046072054531129378
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Epoch 10/15 - loss = 0.024919378473509772
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Epoch 11/15 - loss = 0.014504998499450883
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Epoch 12/15 - loss = 0.0093448161379050161
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Epoch 13/15 - loss = 0.0067649700132868147
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Epoch 14/15 - loss = 0.0054588596501628835
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Epoch 15/15 - loss = 0.0047859930295160499
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[ 2.08995, 0.802632 ]
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gap> theta := SkeletalSmoothMaps.Constant( parameters );
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ℝ^0 -> ℝ^2
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gap> Display( theta );
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ℝ^0 -> ℝ^2
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‣ 2.08995
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‣ 0.802632
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gap> f_theta := ReparametriseMorphism( f, theta );;
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gap> Display( f_theta );
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ℝ^1 -> ℝ^1 defined by:
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Underlying Object:
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-----------------
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ℝ^0
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Underlying Morphism:
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-------------------
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ℝ^1 -> ℝ^1
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‣ 2.08995 * x1 + 0.802632
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gap> f_theta := UnderlyingMorphism( f_theta );
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ℝ^1 -> ℝ^1
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gap> Eval( f_theta, [ 1 ] );
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[ 2.89259 ]
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gap> Eval( f_theta, [ 2 ] );
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[ 4.98254 ]
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gap> Eval( f_theta, [ 3 ] );
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[ 7.07249 ]

‎tst/example-in-readme-2.tst‎

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gap> LoadPackage( "MachineLearningForCAP" );
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true
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gap> Para := CategoryOfParametrisedMorphisms( SkeletalSmoothMaps );
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CategoryOfParametrisedMorphisms( SkeletalSmoothMaps )
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gap> class_1 := Concatenation( List( [ -2 .. 3 ], i -> [ [ i, i - 1, 1, 0, 0 ], [ i + 1, i - 1, 1, 0, 0 ] ] ) );;
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gap> class_2 := Concatenation( List( [ -3 .. -1 ], i -> List( [ i + 1 .. - i - 1 ], j -> [ i, j, 0, 1, 0 ] ) ) );;
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gap> class_3 := Concatenation( List( [ 1 .. 3 ], i -> List( [ - i + 1 .. i - 1 ], j -> [ j, i, 0, 0, 1 ] ) ) );;
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gap> training_set := Concatenation( class_1, class_2, class_3 );
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[ [ -2, -3, 1, 0, 0 ], [ -1, -3, 1, 0, 0 ], [ -1, -2, 1, 0, 0 ], [ 0, -2, 1, 0, 0 ], [ 0, -1, 1, 0, 0 ],
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[ 1, -1, 1, 0, 0 ], [ 1, 0, 1, 0, 0 ], [ 2, 0, 1, 0, 0 ], [ 2, 1, 1, 0, 0 ], [ 3, 1, 1, 0, 0 ],
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[ 3, 2, 1, 0, 0 ], [ 4, 2, 1, 0, 0 ], [ -3, -2, 0, 1, 0 ], [ -3, -1, 0, 1, 0 ], [ -3, 0, 0, 1, 0 ],
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[ -3, 1, 0, 1, 0 ], [ -3, 2, 0, 1, 0 ], [ -2, -1, 0, 1, 0 ], [ -2, 0, 0, 1, 0 ], [ -2, 1, 0, 1, 0 ],
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[ -1, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 1 ], [ -1, 2, 0, 0, 1 ], [ 0, 2, 0, 0, 1 ], [ 1, 2, 0, 0, 1 ],
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[ -2, 3, 0, 0, 1 ], [ -1, 3, 0, 0, 1 ], [ 0, 3, 0, 0, 1 ], [ 1, 3, 0, 0, 1 ], [ 2, 3, 0, 0, 1 ] ]
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gap> input_dim := 2;; output_dim := 3;; hidden_dims := [ ];;
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gap> f := PredictionMorphismOfNeuralNetwork( Para, input_dim, hidden_dims, output_dim, "Softmax" );;
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gap> input := ConvertToExpressions( [ "theta_1", "theta_2", "theta_3", "theta_4", "theta_5", "theta_6", "theta_7", "theta_8", "theta_9", "x1", "x2" ] );;
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gap> Display( f : dummy_input := input );
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ℝ^2 -> ℝ^3 defined by:
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Underlying Object:
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-----------------
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ℝ^9
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Underlying Morphism:
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-------------------
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ℝ^11 -> ℝ^3
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‣ Exp( (theta_1 * x1 + theta_2 * x2 + theta_3) )
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/ (Exp( theta_1 * x1 + theta_2 * x2 + theta_3 ) + Exp( (theta_4 * x1 + theta_5 * x2 + theta_6) ) + Exp( (theta_7 * x1 + theta_8 * x2 + theta_9) ))
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‣ Exp( (theta_4 * x1 + theta_5 * x2 + theta_6) )
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/ (Exp( theta_1 * x1 + theta_2 * x2 + theta_3 ) + Exp( (theta_4 * x1 + theta_5 * x2 + theta_6) ) + Exp( (theta_7 * x1 + theta_8 * x2 + theta_9) ))
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‣ Exp( (theta_7 * x1 + theta_8 * x2 + theta_9) )
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/ (Exp( theta_1 * x1 + theta_2 * x2 + theta_3 ) + Exp( (theta_4 * x1 + theta_5 * x2 + theta_6) ) + Exp( (theta_7 * x1 + theta_8 * x2 + theta_9) ))
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gap> parameters := [ 0.1, -0.1, 0, 0.1, 0.2, 0, -0.2, 0.3, 0 ];;
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gap> x := [ 1, 2 ];;
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gap> prediction_x := Eval( f, [ parameters, x ] );
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[ 0.223672, 0.407556, 0.368772 ]
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gap> PositionMaximum( prediction_x );
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2
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gap> ell := LossMorphismOfNeuralNetwork( Para, input_dim, hidden_dims, output_dim, "Softmax" );;
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gap> input := ConvertToExpressions( [ "theta_1", "theta_2", "theta_3", "theta_4", "theta_5", "theta_6", "theta_7", "theta_8", "theta_9", "x1", "x2", "y1", "y2", "y3" ] );;
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gap> Display( ell : dummy_input := input );
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ℝ^5 -> ℝ^1 defined by:
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Underlying Object:
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-----------------
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ℝ^9
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Underlying Morphism:
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-------------------
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ℝ^14 -> ℝ^1
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‣ ((Log( Exp( theta_1 * x1 + theta_2 * x2 + theta_3 ) + Exp( (theta_4 * x1 + theta_5 * x2 + theta_6) ) + Exp( (theta_7 * x1 + theta_8 * x2 + theta_9) ) ) - (theta_1 * x1 + theta_2 * x2 + theta_3)) * y1
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+ (Log( Exp( theta_1 * x1 + theta_2 * x2 + theta_3 ) + Exp( (theta_4 * x1 + theta_5 * x2 + theta_6) ) + Exp( (theta_7 * x1 + theta_8 * x2 + theta_9) ) ) - (theta_4 * x1 + theta_5 * x2 + theta_6)) * y2
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+ (Log( Exp( theta_1 * x1 + theta_2 * x2 + theta_3 ) + Exp( (theta_4 * x1 + theta_5 * x2 + theta_6) ) + Exp( (theta_7 * x1 + theta_8 * x2 + theta_9) ) ) - (theta_7 * x1 + theta_8 * x2 + theta_9)) * y3) / 3
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gap> Lenses := CategoryOfLenses( SkeletalSmoothMaps );
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CategoryOfLenses( SkeletalSmoothMaps )
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gap> optimizer := Lenses.AdamOptimizer( : learning_rate := 0.01, beta_1 := 0.9, beta_2 := 0.999 );;
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gap> batch_size := 1;;
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gap> one_epoch_update := OneEpochUpdateLens( ell, optimizer, training_set, batch_size );
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(ℝ^28, ℝ^28) -> (ℝ^1, ℝ^0) defined by:
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Get Morphism:
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----------
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ℝ^28 -> ℝ^1
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Put Morphism:
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----------
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ℝ^28 -> ℝ^28
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gap> parameters := [ 0.1, -0.1, 0, 0.1, 0.2, 0, -0.2, 0.3, 0 ];;
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gap> w := Concatenation( [ 1 ], 0 * parameters, 0 * parameters, parameters );;
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gap> Eval( GetMorphism( one_epoch_update ), w );
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[ 0.345836 ]
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gap> new_w := Eval( PutMorphism( one_epoch_update ), w );
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[ 31, 0.104642, -0.355463, -0.197135, -0.109428, -0.147082, 0.00992963,
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0.00478679, 0.502546, 0.187206, 0.0105493, 0.00642903, 0.00211548,
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0.00660062, 0.00274907, 0.00110985, 0.00278786, 0.0065483, 0.00112838,
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5.45195, -1.26208, 3.82578, -5.40639, -0.952146, -3.42835, -2.79496, 3.09008, -6.80672 ]
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gap> nr_epochs := 4;;
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gap> w := Fit( one_epoch_update, nr_epochs, w );
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Epoch 0/4 - loss = 0.34583622811001763
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Epoch 1/4 - loss = 0.6449437167091393
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Epoch 2/4 - loss = 0.023811108587716449
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Epoch 3/4 - loss = 0.0036371652708073405
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Epoch 4/4 - loss = 0.0030655216725219204
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[ 121, -4.57215e-05, -0.00190417, -0.0014116, -0.00181528, 0.00108949, 0.00065518, 0.001861, 0.000814679,
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0.000756424, 0.0104885, 0.00846858, 0.0022682, 0.00784643, 0.00551702, 0.0014626, 0.00351408, 0.00640225,
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0.00115053, 5.09137, -4.83379, 3.06257, -5.70976, 0.837175, -4.23622, -1.71171, 5.54301, -4.80856 ]
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gap> theta := SplitDenseList( w, [ 19, 9 ] )[2];
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[ 5.09137, -4.83379, 3.06257, -5.70976, 0.837175, -4.23622, -1.71171, 5.54301, -4.80856 ]
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gap> theta := SkeletalSmoothMaps.Constant( theta );
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ℝ^0 -> ℝ^9
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gap> f_theta := ReparametriseMorphism( f, theta );;
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gap> Display( f_theta );
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ℝ^2 -> ℝ^3 defined by:
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Underlying Object:
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-----------------
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ℝ^0
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Underlying Morphism:
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-------------------
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ℝ^2 -> ℝ^3
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‣ Exp( (5.09137 * x1 + (- 4.83379) * x2 + 3.06257) )
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/ (Exp( 5.09137 * x1 + (- 4.83379) * x2 + 3.06257 )
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+ Exp( ((- 5.70976) * x1 + 0.837175 * x2 + (- 4.23622)) )
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+ Exp( ((- 1.71171) * x1 + 5.54301 * x2 + (- 4.80856)) ))
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‣ Exp( ((- 5.70976) * x1 + 0.837175 * x2 + (- 4.23622)) )
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/ (Exp( 5.09137 * x1 + (- 4.83379) * x2 + 3.06257 )
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+ Exp( ((- 5.70976) * x1 + 0.837175 * x2 + (- 4.23622)) )
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+ Exp( ((- 1.71171) * x1 + 5.54301 * x2 + (- 4.80856)) ))
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‣ Exp( ((- 1.71171) * x1 + 5.54301 * x2 + (- 4.80856)) )
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/ (Exp( 5.09137 * x1 + (- 4.83379) * x2 + 3.06257 )
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+ Exp( ((- 5.70976) * x1 + 0.837175 * x2 + (- 4.23622)) )
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+ Exp( ((- 1.71171) * x1 + 5.54301 * x2 + (- 4.80856)) ))
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gap> f_theta := UnderlyingMorphism( f_theta );
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ℝ^2 -> ℝ^3
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gap> predictions_vec := Eval( f_theta, [ 1, -1 ] );
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[ 1., 4.74723e-11, 1.31974e-11 ]
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gap> PositionMaximum( predictions_vec );
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1
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gap> predictions_vec := Eval( f_theta, [ 1, 3 ] );
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[ 7.13122e-08, 2.40484e-08, 1. ]
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gap> PositionMaximum( predictions_vec );
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3

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