func GradientDescent(X *mat64.Dense, y *mat64.Vector, alpha, tolerance float64, maxIters int) *mat64.Vector {
// m = Number of Training Examples
// n = Number of Features
m, n := X.Dims()
h := mat64.NewVector(m, nil)
partials := mat64.NewVector(n, nil)
new_theta := mat64.NewVector(n, nil)
Regression:
for i := 0; i < maxIters; i++ {
// Calculate partial derivatives
h.MulVec(X, new_theta)
for el := 0; el < m; el++ {
val := (h.At(el, 0) - y.At(el, 0)) / float64(m)
h.SetVec(el, val)
}
partials.MulVec(X.T(), h)
// Update theta values
for el := 0; el < n; el++ {
new_val := new_theta.At(el, 0) - (alpha * partials.At(el, 0))
new_theta.SetVec(el, new_val)
}
// Check the "distance" to the local minumum
dist := math.Sqrt(mat64.Dot(partials, partials))
if dist <= tolerance {
break Regression
}
}
return new_theta
}
func (self *Layer) Update(learningConfiguration LearningConfiguration) {
var deltas mat64.Dense
deltas.Mul(self.Deltas, self.Input)
rows, cols := self.Weight.Dims()
weight := self.Weight.View(0, 0, rows-1, cols).(*mat64.Dense)
if *learningConfiguration.Decay > 0 {
var decay mat64.Dense
decay.Scale(*learningConfiguration.Decay, weight)
deltas.Sub(&deltas, decay.T())
}
deltas.Scale(*learningConfiguration.Rate, &deltas)
weight.Sub(weight, deltas.T())
}
// Back propagate the error and update the weights.
func (m *Mind) Back(input *mat64.Dense, output *mat64.Dense) {
ErrorOutputLayer := &mat64.Dense{}
DeltaOutputLayer := &mat64.Dense{}
HiddenOutputChanges := &mat64.Dense{}
DeltaHiddenLayer := &mat64.Dense{}
InputHiddenChanges := &mat64.Dense{}
ErrorOutputLayer.Sub(output, m.Results.OutputResult)
DeltaOutputLayer.MulElem(m.ActivatePrime(m.Results.OutputSum), ErrorOutputLayer)
HiddenOutputChanges.Mul(m.Results.HiddenResult.T(), DeltaOutputLayer)
HiddenOutputChanges.Scale(m.LearningRate, HiddenOutputChanges)
m.Weights.HiddenOutput.Add(HiddenOutputChanges, m.Weights.HiddenOutput)
DeltaHiddenLayer.Mul(DeltaOutputLayer, m.Weights.HiddenOutput.T())
DeltaHiddenLayer.MulElem(m.ActivatePrime(m.Results.HiddenSum), DeltaHiddenLayer)
InputHiddenChanges.Mul(input.T(), DeltaHiddenLayer)
InputHiddenChanges.Scale(m.LearningRate, InputHiddenChanges)
m.Weights.InputHidden.Add(InputHiddenChanges, m.Weights.InputHidden)
}
// Back propagate the error and update the weights.
func (m *Mind) Back(input *mat64.Dense, output *mat64.Dense) {
ErrorOutputLayer := mat64.NewDense(1, 1, nil)
ErrorOutputLayer.Sub(output, m.Results.OutputResult)
DeltaOutputLayer := m.ActivatePrime(m.Results.OutputSum)
DeltaOutputLayer.MulElem(DeltaOutputLayer, ErrorOutputLayer)
HiddenOutputChanges := mat64.DenseCopyOf(m.Results.HiddenResult.T())
HiddenOutputChanges.Product(DeltaOutputLayer)
HiddenOutputChanges.Scale(m.LearningRate, HiddenOutputChanges)
m.Weights.HiddenOutput.Add(m.Weights.HiddenOutput, HiddenOutputChanges)
DeltaHiddenLayer := mat64.DenseCopyOf(DeltaOutputLayer)
DeltaHiddenLayer.Product(DeltaOutputLayer, m.Weights.HiddenOutput.T())
DeltaHiddenLayer.MulElem(DeltaHiddenLayer, m.ActivatePrime(m.Results.HiddenSum))
InputHiddenChanges := mat64.DenseCopyOf(input.T())
InputHiddenChanges.Product(DeltaHiddenLayer)
InputHiddenChanges.Scale(m.LearningRate, InputHiddenChanges)
m.Weights.InputHidden.Add(m.Weights.InputHidden, InputHiddenChanges)
}
func TestChebyshev(t *testing.T) {
var vectorX, vectorY *mat64.Dense
chebyshev := NewChebyshev()
Convey("Given two vectors", t, func() {
vectorX = mat64.NewDense(4, 1, []float64{1, 2, 3, 4})
vectorY = mat64.NewDense(4, 1, []float64{-5, -6, 7, 8})
Convey("When calculating distance with two vectors", func() {
result := chebyshev.Distance(vectorX, vectorY)
Convey("The result should be 8", func() {
So(result, ShouldEqual, 8)
})
})
Convey("When calculating distance with row vectors", func() {
vectorX.Copy(vectorX.T())
vectorY.Copy(vectorY.T())
result := chebyshev.Distance(vectorX, vectorY)
Convey("The result should be 8", func() {
So(result, ShouldEqual, 8)
})
})
Convey("When calculating distance with different dimension matrices", func() {
vectorX.Clone(vectorX.T())
So(func() { chebyshev.Distance(vectorX, vectorY) }, ShouldPanic)
})
})
}
func TestCranberrra(t *testing.T) {
var vectorX, vectorY *mat64.Dense
cranberra := NewCranberra()
Convey("Given two vectors that are same", t, func() {
vec := mat64.NewDense(7, 1, []float64{0, 1, -2, 3.4, 5, -6.7, 89})
distance := cranberra.Distance(vec, vec)
Convey("The result should be 0", func() {
So(distance, ShouldEqual, 0)
})
})
Convey("Given two vectors", t, func() {
vectorX = mat64.NewDense(5, 1, []float64{1, 2, 3, 4, 9})
vectorY = mat64.NewDense(5, 1, []float64{-5, -6, 7, 4, 3})
Convey("When calculating distance with two vectors", func() {
result := cranberra.Distance(vectorX, vectorY)
Convey("The result should be 2.9", func() {
So(result, ShouldEqual, 2.9)
})
})
Convey("When calculating distance with row vectors", func() {
vectorX.Copy(vectorX.T())
vectorY.Copy(vectorY.T())
result := cranberra.Distance(vectorX, vectorY)
Convey("The result should be 2.9", func() {
So(result, ShouldEqual, 2.9)
})
})
Convey("When calculating distance with different dimension matrices", func() {
vectorX.Clone(vectorX.T())
So(func() { cranberra.Distance(vectorX, vectorY) }, ShouldPanic)
})
})
}
func TestManhattan(t *testing.T) {
var vectorX, vectorY *mat64.Dense
manhattan := NewManhattan()
Convey("Given two vectors that are same", t, func() {
vec := mat64.NewDense(7, 1, []float64{0, 1, -2, 3.4, 5, -6.7, 89})
distance := manhattan.Distance(vec, vec)
Convey("The result should be 0", func() {
So(distance, ShouldEqual, 0)
})
})
Convey("Given two vectors", t, func() {
vectorX = mat64.NewDense(3, 1, []float64{2, 2, 3})
vectorY = mat64.NewDense(3, 1, []float64{1, 4, 5})
Convey("When calculating distance with column vectors", func() {
result := manhattan.Distance(vectorX, vectorY)
Convey("The result should be 5", func() {
So(result, ShouldEqual, 5)
})
})
Convey("When calculating distance with row vectors", func() {
vectorX.Copy(vectorX.T())
vectorY.Copy(vectorY.T())
result := manhattan.Distance(vectorX, vectorY)
Convey("The result should be 5", func() {
So(result, ShouldEqual, 5)
})
})
Convey("When calculating distance with different dimension matrices", func() {
vectorX.Clone(vectorX.T())
So(func() { manhattan.Distance(vectorX, vectorY) }, ShouldPanic)
})
})
}
// Factors returns matrices W and H that are non-negative factors of V within the
// specified tolerance and computation limits given initial non-negative solutions Wo
// and Ho.
func Factors(V, Wo, Ho *mat64.Dense, c Config) (W, H *mat64.Dense, ok bool) {
to := time.Now()
W = Wo
H = Ho
var (
wr, wc = W.Dims()
hr, hc = H.Dims()
tmp mat64.Dense
)
var vhT mat64.Dense
gW := mat64.NewDense(wr, wc, nil)
tmp.Mul(H, H.T())
gW.Mul(W, &tmp)
vhT.Mul(V, H.T())
gW.Sub(gW, &vhT)
var wTv mat64.Dense
gH := mat64.NewDense(hr, hc, nil)
tmp.Reset()
tmp.Mul(W.T(), W)
gH.Mul(&tmp, H)
wTv.Mul(W.T(), V)
gH.Sub(gH, &wTv)
var gHT, gWHT mat64.Dense
gHT.Clone(gH.T())
gWHT.Stack(gW, &gHT)
grad := mat64.Norm(&gWHT, 2)
tolW := math.Max(0.001, c.Tolerance) * grad
tolH := tolW
var (
_ok bool
iter int
)
decFiltW := func(r, c int, v float64) float64 {
// decFiltW is applied to gW, so v = gW.At(r, c).
if v < 0 || W.At(r, c) > 0 {
return v
}
return 0
}
decFiltH := func(r, c int, v float64) float64 {
// decFiltH is applied to gH, so v = gH.At(r, c).
if v < 0 || H.At(r, c) > 0 {
return v
}
return 0
}
var vT, hT, wT mat64.Dense
for i := 0; i < c.MaxIter; i++ {
gW.Apply(decFiltW, gW)
gH.Apply(decFiltH, gH)
var proj float64
for _, v := range gW.RawMatrix().Data {
proj += v * v
}
for _, v := range gH.RawMatrix().Data {
proj += v * v
}
proj = math.Sqrt(proj)
if proj < c.Tolerance*grad || time.Now().Sub(to) > c.Limit {
break
}
vT.Clone(V.T())
hT.Clone(H.T())
wT.Clone(W.T())
W, gW, iter, ok = nnlsSubproblem(&vT, &hT, &wT, tolW, c.MaxOuterSub, c.MaxInnerSub)
if iter == 0 {
tolW *= 0.1
}
wT.Reset()
wT.Clone(W.T())
W = &wT
var gWT mat64.Dense
gWT.Clone(gW.T())
*gW = gWT
H, gH, iter, _ok = nnlsSubproblem(V, W, H, tolH, c.MaxOuterSub, c.MaxInnerSub)
ok = ok && _ok
if iter == 0 {
tolH *= 0.1
}
}
return W, H, ok
}
func nnlsSubproblem(V, W, Ho *mat64.Dense, tol float64, outer, inner int) (H, G *mat64.Dense, i int, ok bool) {
H = new(mat64.Dense)
H.Clone(Ho)
var WtV, WtW mat64.Dense
WtV.Mul(W.T(), V)
WtW.Mul(W.T(), W)
alpha, beta := 1., 0.1
decFilt := func(r, c int, v float64) float64 {
// decFilt is applied to G, so v = G.At(r, c).
if v < 0 || H.At(r, c) > 0 {
return v
}
return 0
}
G = new(mat64.Dense)
for i = 0; i < outer; i++ {
G.Mul(&WtW, H)
G.Sub(G, &WtV)
G.Apply(decFilt, G)
if mat64.Norm(G, 2) < tol {
break
}
var (
reduce bool
Hp *mat64.Dense
d, dQ mat64.Dense
)
for j := 0; j < inner; j++ {
var Hn mat64.Dense
Hn.Scale(alpha, G)
Hn.Sub(H, &Hn)
Hn.Apply(posFilt, &Hn)
d.Sub(&Hn, H)
dQ.Mul(&WtW, &d)
dQ.MulElem(&dQ, &d)
d.MulElem(G, &d)
sufficient := 0.99*mat64.Sum(&d)+0.5*mat64.Sum(&dQ) < 0
if j == 0 {
reduce = !sufficient
Hp = H
}
if reduce {
if sufficient {
H = &Hn
ok = true
break
} else {
alpha *= beta
}
} else {
if !sufficient || mat64.Equal(Hp, &Hn) {
H = Hp
break
} else {
alpha /= beta
Hp = &Hn
}
}
}
}
return H, G, i, ok
}
func (self *Layer) BackwardOutput(values *mat64.Dense,
error_function ErrorFunction) {
// TODO(ariw): ErrorFunction Delta use goes here.
values.Sub(self.Output, values)
self.Deltas.MulElem(values.T(), self.Derivatives)
}
func simplex(initialBasic []int, c []float64, A mat64.Matrix, b []float64, tol float64) (float64, []float64, []int, error) {
err := verifyInputs(initialBasic, c, A, b)
if err != nil {
if err == ErrUnbounded {
return math.Inf(-1), nil, nil, ErrUnbounded
}
return math.NaN(), nil, nil, err
}
m, n := A.Dims()
// There is at least one optimal solution to the LP which is at the intersection
// to a set of constraint boundaries. For a standard form LP with m variables
// and n equality constraints, at least m-n elements of x must equal zero
// at optimality. The Simplex algorithm solves the standard-form LP by starting
// at an initial constraint vertex and successively moving to adjacent constraint
// vertices. At every vertex, the set of non-zero x values is the "basic
// feasible solution". The list of non-zero x's are maintained in basicIdxs,
// the respective columns of A are in ab, and the actual non-zero values of
// x are in xb.
//
// The LP is equality constrained such that A * x = b. This can be expanded
// to
// ab * xb + an * xn = b
// where ab are the columns of a in the basic set, and an are all of the
// other columns. Since each element of xn is zero by definition, this means
// that for all feasible solutions xb = ab^-1 * b.
//
// Before the simplex algorithm can start, an initial feasible solution must
// be found. If initialBasic is non-nil a feasible solution has been supplied.
// Otherwise the "Phase I" problem must be solved to find an initial feasible
// solution.
var basicIdxs []int // The indices of the non-zero x values.
var ab *mat64.Dense // The subset of columns of A listed in basicIdxs.
var xb []float64 // The non-zero elements of x. xb = ab^-1 b
if initialBasic != nil {
// InitialBasic supplied. Panic if incorrect length or infeasible.
if len(initialBasic) != m {
panic("lp: incorrect number of initial vectors")
}
ab = extractColumns(A, initialBasic)
xb, err = initializeFromBasic(ab, b)
if err != nil {
panic(err)
}
basicIdxs = make([]int, len(initialBasic))
copy(basicIdxs, initialBasic)
} else {
// No inital basis supplied. Solve the PhaseI problem.
basicIdxs, ab, xb, err = findInitialBasic(A, b)
if err != nil {
return math.NaN(), nil, nil, err
}
}
// basicIdxs contains the indexes for an initial feasible solution,
// ab contains the extracted columns of A, and xb contains the feasible
// solution. All x not in the basic set are 0 by construction.
// nonBasicIdx is the set of nonbasic variables.
nonBasicIdx := make([]int, 0, n-m)
inBasic := make(map[int]struct{})
for _, v := range basicIdxs {
inBasic[v] = struct{}{}
}
for i := 0; i < n; i++ {
_, ok := inBasic[i]
if !ok {
nonBasicIdx = append(nonBasicIdx, i)
}
}
// cb is the subset of c for the basic variables. an and cn
// are the equivalents to ab and cb but for the nonbasic variables.
cb := make([]float64, len(basicIdxs))
for i, idx := range basicIdxs {
cb[i] = c[idx]
}
cn := make([]float64, len(nonBasicIdx))
for i, idx := range nonBasicIdx {
cn[i] = c[idx]
}
an := extractColumns(A, nonBasicIdx)
bVec := mat64.NewVector(len(b), b)
cbVec := mat64.NewVector(len(cb), cb)
// Temporary data needed each iteration. (Described later)
r := make([]float64, n-m)
move := make([]float64, m)
// Solve the linear program starting from the initial feasible set. This is
// the "Phase 2" problem.
//
// Algorithm:
// 1) Compute the "reduced costs" for the non-basic variables. The reduced
// costs are the lagrange multipliers of the constraints.
// r = cn - an^T * ab^-T * cb
// 2) If all of the reduced costs are positive, no improvement is possible,
// and the solution is optimal (xn can only increase because of
// non-negativity constraints). Otherwise, the solution can be improved and
// one element will be exchanged in the basic set.
// 3) Choose the x_n with the most negative value of r. Call this value xe.
// This variable will be swapped into the basic set.
// 4) Increase xe until the next constraint boundary is met. This will happen
// when the first element in xb becomes 0. The distance xe can increase before
// a given element in xb becomes negative can be found from
// xb = Ab^-1 b - Ab^-1 An xn
// = Ab^-1 b - Ab^-1 Ae xe
// = bhat + d x_e
// xe = bhat_i / - d_i
// where Ae is the column of A corresponding to xe.
// The constraining basic index is the first index for which this is true,
// so remove the element which is min_i (bhat_i / -d_i), assuming d_i is negative.
// If no d_i is less than 0, then the problem is unbounded.
// 5) If the new xe is 0 (that is, bhat_i == 0), then this location is at
// the intersection of several constraints. Use the Bland rule instead
// of the rule in step 4 to avoid cycling.
for {
// Compute reduced costs -- r = cn - an^T ab^-T cb
var tmp mat64.Vector
err = tmp.SolveVec(ab.T(), cbVec)
if err != nil {
break
}
data := make([]float64, n-m)
tmp2 := mat64.NewVector(n-m, data)
tmp2.MulVec(an.T(), &tmp)
floats.SubTo(r, cn, data)
// Replace the most negative element in the simplex. If there are no
// negative entries then the optimal solution has been found.
minIdx := floats.MinIdx(r)
if r[minIdx] >= -tol {
break
}
for i, v := range r {
if math.Abs(v) < rRoundTol {
r[i] = 0
}
}
// Compute the moving distance.
err = computeMove(move, minIdx, A, ab, xb, nonBasicIdx)
if err != nil {
if err == ErrUnbounded {
return math.Inf(-1), nil, nil, ErrUnbounded
}
break
}
// Replace the basic index along the tightest constraint.
replace := floats.MinIdx(move)
if move[replace] <= 0 {
replace, minIdx, err = replaceBland(A, ab, xb, basicIdxs, nonBasicIdx, r, move)
if err != nil {
if err == ErrUnbounded {
return math.Inf(-1), nil, nil, ErrUnbounded
}
break
}
}
// Replace the constrained basicIdx with the newIdx.
basicIdxs[replace], nonBasicIdx[minIdx] = nonBasicIdx[minIdx], basicIdxs[replace]
cb[replace], cn[minIdx] = cn[minIdx], cb[replace]
tmpCol1 := mat64.Col(nil, replace, ab)
tmpCol2 := mat64.Col(nil, minIdx, an)
ab.SetCol(replace, tmpCol2)
an.SetCol(minIdx, tmpCol1)
// Compute the new xb.
xbVec := mat64.NewVector(len(xb), xb)
err = xbVec.SolveVec(ab, bVec)
if err != nil {
break
}
}
// Found the optimum successfully or died trying. The basic variables get
// their values, and the non-basic variables are all zero.
opt := floats.Dot(cb, xb)
xopt := make([]float64, n)
for i, v := range basicIdxs {
xopt[v] = xb[i]
}
return opt, xopt, basicIdxs, err
}
func (lr *LinearRegression) Fit(inst base.FixedDataGrid) error {
// Retrieve row size
_, rows := inst.Size()
// Validate class Attribute count
classAttrs := inst.AllClassAttributes()
if len(classAttrs) != 1 {
return fmt.Errorf("Only 1 class variable is permitted")
}
classAttrSpecs := base.ResolveAttributes(inst, classAttrs)
// Retrieve relevant Attributes
allAttrs := base.NonClassAttributes(inst)
attrs := make([]base.Attribute, 0)
for _, a := range allAttrs {
if _, ok := a.(*base.FloatAttribute); ok {
attrs = append(attrs, a)
}
}
cols := len(attrs) + 1
if rows < cols {
return NotEnoughDataError
}
// Retrieve relevant Attribute specifications
attrSpecs := base.ResolveAttributes(inst, attrs)
// Split into two matrices, observed results (dependent variable y)
// and the explanatory variables (X) - see http://en.wikipedia.org/wiki/Linear_regression
observed := mat64.NewDense(rows, 1, nil)
explVariables := mat64.NewDense(rows, cols, nil)
// Build the observed matrix
inst.MapOverRows(classAttrSpecs, func(row [][]byte, i int) (bool, error) {
val := base.UnpackBytesToFloat(row[0])
observed.Set(i, 0, val)
return true, nil
})
// Build the explainatory variables
inst.MapOverRows(attrSpecs, func(row [][]byte, i int) (bool, error) {
// Set intercepts to 1.0
explVariables.Set(i, 0, 1.0)
for j, r := range row {
explVariables.Set(i, j+1, base.UnpackBytesToFloat(r))
}
return true, nil
})
n := cols
qr := new(mat64.QR)
qr.Factorize(explVariables)
var q, reg mat64.Dense
q.QFromQR(qr)
reg.RFromQR(qr)
var transposed, qty mat64.Dense
transposed.Clone(q.T())
qty.Mul(&transposed, observed)
regressionCoefficients := make([]float64, n)
for i := n - 1; i >= 0; i-- {
regressionCoefficients[i] = qty.At(i, 0)
for j := i + 1; j < n; j++ {
regressionCoefficients[i] -= regressionCoefficients[j] * reg.At(i, j)
}
regressionCoefficients[i] /= reg.At(i, i)
}
lr.disturbance = regressionCoefficients[0]
lr.regressionCoefficients = regressionCoefficients[1:]
lr.fitted = true
lr.attrs = attrs
lr.cls = classAttrs[0]
return nil
}