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Merge pull request #461 from ivandev0/kylchik/jacobi

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Jacobi eigenvalue algorithm
This commit is contained in:
Alexander Nozik 2022-02-20 10:18:06 +03:00 committed by GitHub
commit c80f70fe0f
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4 changed files with 192 additions and 18 deletions
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5
benchmarks/build.gradle.kts
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@ -124,6 +124,11 @@ benchmark {
include("JafamaBenchmark")
}
configurations.register("tensorAlgebra") {
commonConfiguration()
include("TensorAlgebraBenchmark")
}
configurations.register("viktor") {
commonConfiguration()
include("ViktorBenchmark")

12
benchmarks/src/jvmMain/kotlin/space/kscience/kmath/benchmarks/DotBenchmark.kt
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@ -15,10 +15,8 @@ import space.kscience.kmath.linear.invoke
import space.kscience.kmath.linear.linearSpace
import space.kscience.kmath.multik.multikAlgebra
import space.kscience.kmath.operations.DoubleField
import space.kscience.kmath.operations.invoke
import space.kscience.kmath.structures.Buffer
import space.kscience.kmath.tensorflow.produceWithTF
import space.kscience.kmath.tensors.core.DoubleTensorAlgebra
import space.kscience.kmath.tensors.core.tensorAlgebra
import kotlin.random.Random
@ -36,9 +34,6 @@ internal class DotBenchmark {
random.nextDouble()
}
val tensor1 = DoubleTensorAlgebra.randomNormal(shape = intArrayOf(dim, dim), 12224)
val tensor2 = DoubleTensorAlgebra.randomNormal(shape = intArrayOf(dim, dim), 12225)
val cmMatrix1 = CMLinearSpace { matrix1.toCM() }
val cmMatrix2 = CMLinearSpace { matrix2.toCM() }
@ -51,7 +46,7 @@ internal class DotBenchmark {
fun tfDot(blackhole: Blackhole) {
blackhole.consume(
DoubleField.produceWithTF {
tensor1 dot tensor2
matrix1 dot matrix1
}
)
}
@ -95,9 +90,4 @@ internal class DotBenchmark {
fun doubleDot(blackhole: Blackhole) = with(DoubleField.linearSpace) {
blackhole.consume(matrix1 dot matrix2)
}
@Benchmark
fun doubleTensorDot(blackhole: Blackhole) = DoubleTensorAlgebra.invoke {
blackhole.consume(tensor1 dot tensor2)
}
}

37
benchmarks/src/jvmMain/kotlin/space/kscience/kmath/benchmarks/TensorAlgebraBenchmark.kt Normal file
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@ -0,0 +1,37 @@
/*
* Copyright 2018-2021 KMath contributors.
* Use of this source code is governed by the Apache 2.0 license that can be found in the license/LICENSE.txt file.
*/
package space.kscience.kmath.benchmarks
import kotlinx.benchmark.Benchmark
import kotlinx.benchmark.Blackhole
import kotlinx.benchmark.Scope
import kotlinx.benchmark.State
import space.kscience.kmath.linear.linearSpace
import space.kscience.kmath.linear.matrix
import space.kscience.kmath.linear.symmetric
import space.kscience.kmath.operations.DoubleField
import space.kscience.kmath.tensors.core.tensorAlgebra
import kotlin.random.Random
@State(Scope.Benchmark)
internal class TensorAlgebraBenchmark {
companion object {
private val random = Random(12224)
private const val dim = 30
private val matrix = DoubleField.linearSpace.matrix(dim, dim).symmetric { _, _ -> random.nextDouble() }
}
@Benchmark
fun tensorSymEigSvd(blackhole: Blackhole) = with(Double.tensorAlgebra) {
blackhole.consume(matrix.symEigSvd(1e-10))
}
@Benchmark
fun tensorSymEigJacobi(blackhole: Blackhole) = with(Double.tensorAlgebra) {
blackhole.consume(matrix.symEigJacobi(50, 1e-10))
}
}

154
kmath-tensors/src/commonMain/kotlin/space/kscience/kmath/tensors/core/DoubleTensorAlgebra.kt
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@ -9,10 +9,7 @@
package space.kscience.kmath.tensors.core
import space.kscience.kmath.misc.PerformancePitfall
import space.kscience.kmath.nd.MutableStructure2D
import space.kscience.kmath.nd.StructureND
import space.kscience.kmath.nd.as1D
import space.kscience.kmath.nd.as2D
import space.kscience.kmath.nd.*
import space.kscience.kmath.operations.DoubleField
import space.kscience.kmath.structures.MutableBuffer
import space.kscience.kmath.structures.indices
@ -885,7 +882,7 @@ public open class DoubleTensorAlgebra :
return Triple(uTensor.transpose(), sTensor, vTensor.transpose())
}
override fun StructureND<Double>.symEig(): Pair<DoubleTensor, DoubleTensor> = symEig(epsilon = 1e-15)
override fun StructureND<Double>.symEig(): Pair<DoubleTensor, DoubleTensor> = symEigJacobi(maxIteration = 50, epsilon = 1e-15)
/**
* Returns eigenvalues and eigenvectors of a real symmetric matrix input or a batch of real symmetric matrices,
@ -895,7 +892,7 @@ public open class DoubleTensorAlgebra :
* and when the cosine approaches 1 in the SVD algorithm.
* @return a pair `eigenvalues to eigenvectors`.
*/
public fun StructureND<Double>.symEig(epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
public fun StructureND<Double>.symEigSvd(epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
checkSymmetric(tensor, epsilon)
fun MutableStructure2D<Double>.cleanSym(n: Int) {
@ -922,6 +919,151 @@ public open class DoubleTensorAlgebra :
return eig to v
}
public fun StructureND<Double>.symEigJacobi(maxIteration: Int, epsilon: Double): Pair<DoubleTensor, DoubleTensor> {
checkSymmetric(tensor, epsilon)
val size = this.dimension
val eigenvectors = zeros(this.shape)
val eigenvalues = zeros(this.shape.sliceArray(0 until size - 1))
var eigenvalueStart = 0
var eigenvectorStart = 0
for (matrix in tensor.matrixSequence()) {
val matrix2D = matrix.as2D()
val (d, v) = matrix2D.jacobiHelper(maxIteration, epsilon)
for (i in 0 until matrix2D.rowNum) {
for (j in 0 until matrix2D.colNum) {
eigenvectors.mutableBuffer.array()[eigenvectorStart + i * matrix2D.rowNum + j] = v[i, j]
}
}
for (i in 0 until matrix2D.rowNum) {
eigenvalues.mutableBuffer.array()[eigenvalueStart + i] = d[i]
}
eigenvalueStart += this.shape.last()
eigenvectorStart += this.shape.last() * this.shape.last()
}
return eigenvalues to eigenvectors
}
private fun MutableStructure2D<Double>.jacobiHelper(
maxIteration: Int,
epsilon: Double
): Pair<Structure1D<Double>, Structure2D<Double>> {
val n = this.shape[0]
val A_ = this.copy()
val V = eye(n)
val D = DoubleTensor(intArrayOf(n), (0 until this.rowNum).map { this[it, it] }.toDoubleArray()).as1D()
val B = DoubleTensor(intArrayOf(n), (0 until this.rowNum).map { this[it, it] }.toDoubleArray()).as1D()
val Z = zeros(intArrayOf(n)).as1D()
// assume that buffered tensor is square matrix
operator fun BufferedTensor<Double>.get(i: Int, j: Int): Double {
return this.mutableBuffer.array()[bufferStart + i * this.shape[0] + j]
}
operator fun BufferedTensor<Double>.set(i: Int, j: Int, value: Double) {
this.mutableBuffer.array()[bufferStart + i * this.shape[0] + j] = value
}
fun maxOffDiagonal(matrix: BufferedTensor<Double>): Double {
var maxOffDiagonalElement = 0.0
for (i in 0 until n - 1) {
for (j in i + 1 until n) {
maxOffDiagonalElement = max(maxOffDiagonalElement, abs(matrix[i, j]))
}
}
return maxOffDiagonalElement
}
fun rotate(a: BufferedTensor<Double>, s: Double, tau: Double, i: Int, j: Int, k: Int, l: Int) {
val g = a[i, j]
val h = a[k, l]
a[i, j] = g - s * (h + g * tau)
a[k, l] = h + s * (g - h * tau)
}
fun jacobiIteration(
a: BufferedTensor<Double>,
v: BufferedTensor<Double>,
d: MutableStructure1D<Double>,
z: MutableStructure1D<Double>,
) {
for (ip in 0 until n - 1) {
for (iq in ip + 1 until n) {
val g = 100.0 * abs(a[ip, iq])
if (g <= epsilon * abs(d[ip]) && g <= epsilon * abs(d[iq])) {
a[ip, iq] = 0.0
continue
}
var h = d[iq] - d[ip]
val t = when {
g <= epsilon * abs(h) -> (a[ip, iq]) / h
else -> {
val theta = 0.5 * h / (a[ip, iq])
val denominator = abs(theta) + sqrt(1.0 + theta * theta)
if (theta < 0.0) -1.0 / denominator else 1.0 / denominator
}
}
val c = 1.0 / sqrt(1 + t * t)
val s = t * c
val tau = s / (1.0 + c)
h = t * a[ip, iq]
z[ip] -= h
z[iq] += h
d[ip] -= h
d[iq] += h
a[ip, iq] = 0.0
for (j in 0 until ip) {
rotate(a, s, tau, j, ip, j, iq)
}
for (j in (ip + 1) until iq) {
rotate(a, s, tau, ip, j, j, iq)
}
for (j in (iq + 1) until n) {
rotate(a, s, tau, ip, j, iq, j)
}
for (j in 0 until n) {
rotate(v, s, tau, j, ip, j, iq)
}
}
}
}
fun updateDiagonal(
d: MutableStructure1D<Double>,
z: MutableStructure1D<Double>,
b: MutableStructure1D<Double>,
) {
for (ip in 0 until d.size) {
b[ip] += z[ip]
d[ip] = b[ip]
z[ip] = 0.0
}
}
var sm = maxOffDiagonal(A_)
for (iteration in 0 until maxIteration) {
if (sm < epsilon) {
break
}
jacobiIteration(A_, V, D, Z)
updateDiagonal(D, Z, B)
sm = maxOffDiagonal(A_)
}
// TODO sort eigenvalues
return D to V.as2D()
}
/**
* Computes the determinant of a square matrix input, or of each square matrix in a batched input
* using LU factorization algorithm.