From b13765ec1928e0f2b2eb6165a8a5a0e1a057b08f Mon Sep 17 00:00:00 2001 From: Ivan Kylchik Date: Sun, 20 Feb 2022 02:21:52 +0300 Subject: [PATCH] Implement much faster Jacobi algorithm --- benchmarks/build.gradle.kts | 5 + .../benchmarks/TensorAlgebraBenchmark.kt | 37 ++++ .../kmath/tensors/core/DoubleTensorAlgebra.kt | 182 ++++++++++-------- 3 files changed, 149 insertions(+), 75 deletions(-) create mode 100644 benchmarks/src/jvmMain/kotlin/space/kscience/kmath/benchmarks/TensorAlgebraBenchmark.kt diff --git a/benchmarks/build.gradle.kts b/benchmarks/build.gradle.kts index e46f4a9b4..f8d39b9c5 100644 --- a/benchmarks/build.gradle.kts +++ b/benchmarks/build.gradle.kts @@ -124,6 +124,11 @@ benchmark { include("JafamaBenchmark") } + configurations.register("tensorAlgebra") { + commonConfiguration() + include("TensorAlgebraBenchmark") + } + configurations.register("viktor") { commonConfiguration() include("ViktorBenchmark") diff --git a/benchmarks/src/jvmMain/kotlin/space/kscience/kmath/benchmarks/TensorAlgebraBenchmark.kt b/benchmarks/src/jvmMain/kotlin/space/kscience/kmath/benchmarks/TensorAlgebraBenchmark.kt new file mode 100644 index 000000000..38e064e53 --- /dev/null +++ b/benchmarks/src/jvmMain/kotlin/space/kscience/kmath/benchmarks/TensorAlgebraBenchmark.kt @@ -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)) + } +} \ No newline at end of file diff --git a/kmath-tensors/src/commonMain/kotlin/space/kscience/kmath/tensors/core/DoubleTensorAlgebra.kt b/kmath-tensors/src/commonMain/kotlin/space/kscience/kmath/tensors/core/DoubleTensorAlgebra.kt index 7fa6e885d..675ff4191 100644 --- a/kmath-tensors/src/commonMain/kotlin/space/kscience/kmath/tensors/core/DoubleTensorAlgebra.kt +++ b/kmath-tensors/src/commonMain/kotlin/space/kscience/kmath/tensors/core/DoubleTensorAlgebra.kt @@ -9,11 +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.Structure2D -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 @@ -886,7 +882,7 @@ public open class DoubleTensorAlgebra : return Triple(uTensor.transpose(), sTensor, vTensor.transpose()) } - override fun StructureND.symEig(): Pair = symEigJacobi(epsilon = 1e-10) + override fun StructureND.symEig(): Pair = symEigJacobi(maxIteration = 50, epsilon = 1e-15) /** * Returns eigenvalues and eigenvectors of a real symmetric matrix input or a batch of real symmetric matrices, @@ -923,103 +919,139 @@ public open class DoubleTensorAlgebra : return eig to v } - public fun StructureND.symEigJacobi(epsilon: Double): Pair { + public fun StructureND.symEigJacobi(maxIteration: Int, epsilon: Double): Pair { checkSymmetric(tensor, epsilon) val size = this.dimension - val s = zeros(this.shape) + 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()) { - matrix.as2D().jacobiHelper(eigenvalues, s, eigenvalueStart, eigenvectorStart, epsilon) + 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() } - // TODO sort eigenvalues - return eigenvalues to s + return eigenvalues to eigenvectors } private fun MutableStructure2D.jacobiHelper( - eigenvalues: DoubleTensor, - eigenvectors: DoubleTensor, - eigenvalueStart: Int, - eigenvectorStart: Int, + maxIteration: Int, epsilon: Double - ) { - var d = this - var s = eye(this.shape[0]) + ): Pair, Structure2D> { + val A_ = this.copy().as2D() + val V = eye(this.shape[0]).as2D() + val D = DoubleTensor(intArrayOf(this.shape[0]), (0 until this.rowNum).map { this[it, it] }.toDoubleArray()).as1D() + val B = DoubleTensor(intArrayOf(this.shape[0]), (0 until this.rowNum).map { this[it, it] }.toDoubleArray()).as1D() + val Z = zeros(intArrayOf(this.shape[0])).as1D() - // TODO implement cyclic method - do { - // 1. Find max element by abs value that is not on diagonal + fun maxOffDiagonal(matrix: MutableStructure2D): Double { var maxOffDiagonalElement = 0.0 - var maxElementIndex = Pair(0, 0) - for (i in 0 until this.rowNum) { - for (j in 0 until this.colNum) { - if (i == j) continue - if (abs(d[i, j]) > maxOffDiagonalElement) { - maxOffDiagonalElement = abs(d[i, j]) - maxElementIndex = i to j + for (i in 0 until matrix.rowNum - 1) { + for (j in i + 1 until matrix.colNum) { + maxOffDiagonalElement = max(maxOffDiagonalElement, abs(matrix[i, j])) + } + } + return maxOffDiagonalElement + } + + fun rotate(a: MutableStructure2D, 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: MutableStructure2D, + v: MutableStructure2D, + d: MutableStructure1D, + z: MutableStructure1D, + ) { + for (ip in 0 until a.rowNum - 1) { + for (iq in ip + 1 until a.colNum) { + 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 a.rowNum) { + rotate(a, s, tau, ip, j, iq, j) + } + for (j in 0 until a.rowNum) { + rotate(v, s, tau, j, ip, j, iq) } } } + } - // 2. Evaluate "rotation" angle - val dIJ = d[maxElementIndex.first, maxElementIndex.second] - val dII = d[maxElementIndex.first, maxElementIndex.first] - val dJJ = d[maxElementIndex.second, maxElementIndex.second] - - val angle = if (dII == dJJ) { - if (dIJ > 0) PI / 4 else -PI / 4 - } else { - 0.5 * atan(2 * dIJ / (dJJ - dII)) - } - - // 3. Build rotation tensor `s1` - val s1 = eye(this.rowNum) - for (i in 0 until this.rowNum) { - for (j in 0 until this.colNum) { - s1.mutableBuffer.array()[i * this.rowNum + j] = when { - maxElementIndex.first == i && maxElementIndex.first == j -> cos(angle) - maxElementIndex.second == i && maxElementIndex.second == j -> cos(angle) - maxElementIndex.first == i && maxElementIndex.second == j -> sin(angle) - maxElementIndex.first == j && maxElementIndex.second == i -> -sin(angle) - else -> s1.mutableBuffer.array()[i * this.rowNum + j] - } - } - } - - // 4. Evaluate new tensor - d = ((s1.transpose() dot d) dot s1).as2D() - s = s dot s1 - if (d.isDiagonal(epsilon)) break - } while(true) - - // 5. Copy result - for (i in 0 until this.rowNum) { - for (j in 0 until this.colNum) { - eigenvectors.mutableBuffer.array()[eigenvectorStart + i * this.rowNum + j] = s.mutableBuffer.array()[i * this.rowNum + j] + fun updateDiagonal( + d: MutableStructure1D, + z: MutableStructure1D, + b: MutableStructure1D, + ) { + for (ip in 0 until d.size) { + b[ip] += z[ip] + d[ip] = b[ip] + z[ip] = 0.0 } } - for (i in 0 until this.rowNum) { - eigenvalues.mutableBuffer.array()[eigenvalueStart + i] = d[i, i] - } - } - - public fun Structure2D.isDiagonal(epsilon: Double = 1e-9): Boolean { - for (i in 0 until this.rowNum) { - for (j in 0 until this.colNum) { - if (i != j && abs(this[i, j]) > epsilon) { - return false - } + 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_) } - return true + // TODO sort eigenvalues + return D to V } /**