forked from kscience/kmath
Implement much faster Jacobi algorithm
This commit is contained in:
Ivan Kylchik
parent
7aff774bc1
commit
b13765ec19
@ -124,6 +124,11 @@ benchmark {
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include("JafamaBenchmark")
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}
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configurations.register("tensorAlgebra") {
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commonConfiguration()
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include("TensorAlgebraBenchmark")
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}
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configurations.register("viktor") {
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commonConfiguration()
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include("ViktorBenchmark")
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@ -0,0 +1,37 @@
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/*
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* Copyright 2018-2021 KMath contributors.
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* Use of this source code is governed by the Apache 2.0 license that can be found in the license/LICENSE.txt file.
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*/
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package space.kscience.kmath.benchmarks
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import kotlinx.benchmark.Benchmark
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import kotlinx.benchmark.Blackhole
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import kotlinx.benchmark.Scope
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import kotlinx.benchmark.State
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import space.kscience.kmath.linear.linearSpace
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import space.kscience.kmath.linear.matrix
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import space.kscience.kmath.linear.symmetric
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import space.kscience.kmath.operations.DoubleField
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import space.kscience.kmath.tensors.core.tensorAlgebra
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import kotlin.random.Random
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@State(Scope.Benchmark)
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internal class TensorAlgebraBenchmark {
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companion object {
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private val random = Random(12224)
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private const val dim = 30
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private val matrix = DoubleField.linearSpace.matrix(dim, dim).symmetric { _, _ -> random.nextDouble() }
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}
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@Benchmark
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fun tensorSymEigSvd(blackhole: Blackhole) = with(Double.tensorAlgebra) {
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blackhole.consume(matrix.symEigSvd(1e-10))
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}
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@Benchmark
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fun tensorSymEigJacobi(blackhole: Blackhole) = with(Double.tensorAlgebra) {
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blackhole.consume(matrix.symEigJacobi(50, 1e-10))
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}
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}
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@ -9,11 +9,7 @@
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package space.kscience.kmath.tensors.core
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import space.kscience.kmath.misc.PerformancePitfall
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import space.kscience.kmath.nd.MutableStructure2D
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import space.kscience.kmath.nd.Structure2D
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import space.kscience.kmath.nd.StructureND
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import space.kscience.kmath.nd.as1D
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import space.kscience.kmath.nd.as2D
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import space.kscience.kmath.nd.*
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import space.kscience.kmath.operations.DoubleField
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import space.kscience.kmath.structures.MutableBuffer
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import space.kscience.kmath.structures.indices
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@ -886,7 +882,7 @@ public open class DoubleTensorAlgebra :
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return Triple(uTensor.transpose(), sTensor, vTensor.transpose())
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}
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override fun StructureND<Double>.symEig(): Pair<DoubleTensor, DoubleTensor> = symEigJacobi(epsilon = 1e-10)
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override fun StructureND<Double>.symEig(): Pair<DoubleTensor, DoubleTensor> = symEigJacobi(maxIteration = 50, epsilon = 1e-15)
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/**
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* Returns eigenvalues and eigenvectors of a real symmetric matrix input or a batch of real symmetric matrices,
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@ -923,103 +919,139 @@ public open class DoubleTensorAlgebra :
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return eig to v
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}
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public fun StructureND<Double>.symEigJacobi(epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
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public fun StructureND<Double>.symEigJacobi(maxIteration: Int, epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
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checkSymmetric(tensor, epsilon)
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val size = this.dimension
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val s = zeros(this.shape)
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val eigenvectors = zeros(this.shape)
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val eigenvalues = zeros(this.shape.sliceArray(0 until size - 1))
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var eigenvalueStart = 0
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var eigenvectorStart = 0
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for (matrix in tensor.matrixSequence()) {
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matrix.as2D().jacobiHelper(eigenvalues, s, eigenvalueStart, eigenvectorStart, epsilon)
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val matrix2D = matrix.as2D()
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val (d, v) = matrix2D.jacobiHelper(maxIteration, epsilon)
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for (i in 0 until matrix2D.rowNum) {
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for (j in 0 until matrix2D.colNum) {
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eigenvectors.mutableBuffer.array()[eigenvectorStart + i * matrix2D.rowNum + j] = v[i, j]
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}
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}
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for (i in 0 until matrix2D.rowNum) {
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eigenvalues.mutableBuffer.array()[eigenvalueStart + i] = d[i]
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}
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eigenvalueStart += this.shape.last()
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eigenvectorStart += this.shape.last() * this.shape.last()
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}
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// TODO sort eigenvalues
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return eigenvalues to s
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return eigenvalues to eigenvectors
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}
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private fun MutableStructure2D<Double>.jacobiHelper(
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eigenvalues: DoubleTensor,
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eigenvectors: DoubleTensor,
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eigenvalueStart: Int,
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eigenvectorStart: Int,
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maxIteration: Int,
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epsilon: Double
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) {
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var d = this
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var s = eye(this.shape[0])
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): Pair<Structure1D<Double>, Structure2D<Double>> {
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val A_ = this.copy().as2D()
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val V = eye(this.shape[0]).as2D()
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val D = DoubleTensor(intArrayOf(this.shape[0]), (0 until this.rowNum).map { this[it, it] }.toDoubleArray()).as1D()
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val B = DoubleTensor(intArrayOf(this.shape[0]), (0 until this.rowNum).map { this[it, it] }.toDoubleArray()).as1D()
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val Z = zeros(intArrayOf(this.shape[0])).as1D()
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// TODO implement cyclic method
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do {
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// 1. Find max element by abs value that is not on diagonal
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fun maxOffDiagonal(matrix: MutableStructure2D<Double>): Double {
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var maxOffDiagonalElement = 0.0
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var maxElementIndex = Pair(0, 0)
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for (i in 0 until this.rowNum) {
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for (j in 0 until this.colNum) {
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if (i == j) continue
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if (abs(d[i, j]) > maxOffDiagonalElement) {
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maxOffDiagonalElement = abs(d[i, j])
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maxElementIndex = i to j
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for (i in 0 until matrix.rowNum - 1) {
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for (j in i + 1 until matrix.colNum) {
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maxOffDiagonalElement = max(maxOffDiagonalElement, abs(matrix[i, j]))
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}
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}
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return maxOffDiagonalElement
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}
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fun rotate(a: MutableStructure2D<Double>, s: Double, tau: Double, i: Int, j: Int, k: Int, l: Int) {
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val g = a[i, j]
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val h = a[k, l]
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a[i, j] = g - s * (h + g * tau)
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a[k, l] = h + s * (g - h * tau)
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}
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fun jacobiIteration(
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a: MutableStructure2D<Double>,
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v: MutableStructure2D<Double>,
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d: MutableStructure1D<Double>,
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z: MutableStructure1D<Double>,
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) {
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for (ip in 0 until a.rowNum - 1) {
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for (iq in ip + 1 until a.colNum) {
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val g = 100.0 * abs(a[ip, iq])
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if (g <= epsilon * abs(d[ip]) && g <= epsilon * abs(d[iq])) {
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a[ip, iq] = 0.0
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continue
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}
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var h = d[iq] - d[ip]
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val t = when {
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g <= epsilon * abs(h) -> (a[ip, iq]) / h
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else -> {
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val theta = 0.5 * h / (a[ip, iq])
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val denominator = abs(theta) + sqrt(1.0 + theta * theta)
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if (theta < 0.0) -1.0 / denominator else 1.0 / denominator
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}
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}
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val c = 1.0 / sqrt(1 + t * t)
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val s = t * c
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val tau = s / (1.0 + c)
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h = t * a[ip, iq]
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z[ip] -= h
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z[iq] += h
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d[ip] -= h
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d[iq] += h
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a[ip, iq] = 0.0
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for (j in 0 until ip) {
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rotate(a, s, tau, j, ip, j, iq)
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}
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for (j in (ip + 1) until iq) {
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rotate(a, s, tau, ip, j, j, iq)
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}
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for (j in (iq + 1) until a.rowNum) {
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rotate(a, s, tau, ip, j, iq, j)
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}
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for (j in 0 until a.rowNum) {
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rotate(v, s, tau, j, ip, j, iq)
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}
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}
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}
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}
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// 2. Evaluate "rotation" angle
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val dIJ = d[maxElementIndex.first, maxElementIndex.second]
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val dII = d[maxElementIndex.first, maxElementIndex.first]
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val dJJ = d[maxElementIndex.second, maxElementIndex.second]
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val angle = if (dII == dJJ) {
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if (dIJ > 0) PI / 4 else -PI / 4
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} else {
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0.5 * atan(2 * dIJ / (dJJ - dII))
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}
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// 3. Build rotation tensor `s1`
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val s1 = eye(this.rowNum)
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for (i in 0 until this.rowNum) {
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for (j in 0 until this.colNum) {
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s1.mutableBuffer.array()[i * this.rowNum + j] = when {
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maxElementIndex.first == i && maxElementIndex.first == j -> cos(angle)
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maxElementIndex.second == i && maxElementIndex.second == j -> cos(angle)
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maxElementIndex.first == i && maxElementIndex.second == j -> sin(angle)
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maxElementIndex.first == j && maxElementIndex.second == i -> -sin(angle)
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else -> s1.mutableBuffer.array()[i * this.rowNum + j]
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}
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}
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}
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// 4. Evaluate new tensor
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d = ((s1.transpose() dot d) dot s1).as2D()
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s = s dot s1
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if (d.isDiagonal(epsilon)) break
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} while(true)
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// 5. Copy result
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for (i in 0 until this.rowNum) {
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for (j in 0 until this.colNum) {
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eigenvectors.mutableBuffer.array()[eigenvectorStart + i * this.rowNum + j] = s.mutableBuffer.array()[i * this.rowNum + j]
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fun updateDiagonal(
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d: MutableStructure1D<Double>,
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z: MutableStructure1D<Double>,
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b: MutableStructure1D<Double>,
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) {
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for (ip in 0 until d.size) {
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b[ip] += z[ip]
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d[ip] = b[ip]
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z[ip] = 0.0
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}
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}
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for (i in 0 until this.rowNum) {
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eigenvalues.mutableBuffer.array()[eigenvalueStart + i] = d[i, i]
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}
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}
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public fun Structure2D<Double>.isDiagonal(epsilon: Double = 1e-9): Boolean {
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for (i in 0 until this.rowNum) {
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for (j in 0 until this.colNum) {
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if (i != j && abs(this[i, j]) > epsilon) {
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return false
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}
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var sm = maxOffDiagonal(A_)
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for (iteration in 0 until maxIteration) {
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if (sm < epsilon) {
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break
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}
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jacobiIteration(A_, V, D, Z)
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updateDiagonal(D, Z, B)
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sm = maxOffDiagonal(A_)
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}
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return true
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// TODO sort eigenvalues
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return D to V
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}
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/**
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