@tensorflow/tfjs-core
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Hardware-accelerated JavaScript library for machine intelligence
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JavaScript
"use strict";
/**
* @license
* Copyright 2018 Google LLC. All Rights Reserved.
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
* =============================================================================
*/
Object.defineProperty(exports, "__esModule", { value: true });
/**
* Linear algebra ops.
*/
var engine_1 = require("../engine");
var globals_1 = require("../globals");
var tensor_util_env_1 = require("../tensor_util_env");
var util_1 = require("../util");
var array_ops_1 = require("./array_ops");
var binary_ops_1 = require("./binary_ops");
var concat_split_1 = require("./concat_split");
var logical_ops_1 = require("./logical_ops");
var norm_1 = require("./norm");
var operation_1 = require("./operation");
var reduction_ops_1 = require("./reduction_ops");
var tensor_ops_1 = require("./tensor_ops");
/**
* Copy a tensor setting everything outside a central band in each innermost
* matrix to zero.
*
* The band part is computed as follows: Assume input has `k` dimensions
* `[I, J, K, ..., M, N]`, then the output is a tensor with the same shape where
* `band[i, j, k, ..., m, n] = in_band(m, n) * input[i, j, k, ..., m, n]`.
* The indicator function
* `in_band(m, n) = (num_lower < 0 || (m-n) <= num_lower))`
* `&& (num_upper < 0 || (n-m) <= num_upper)`
*
* ```js
* const x = tf.tensor2d([[ 0, 1, 2, 3],
* [-1, 0, 1, 2],
* [-2, -1, 0, 1],
* [-3, -2, -1, 0]]);
* let y = tf.linalg.bandPart(x, 1, -1);
* y.print(); // [[ 0, 1, 2, 3],
* // [-1, 0, 1, 2],
* // [ 0, -1, 0, 1],
* // [ 0, 0 , -1, 0]]
* let z = tf.linalg.bandPart(x, 2, 1);
* z.print(); // [[ 0, 1, 0, 0],
* // [-1, 0, 1, 0],
* // [-2, -1, 0, 1],
* // [ 0, -2, -1, 0]]
* ```
*
* @param x Rank `k` tensor
* @param numLower Number of subdiagonals to keep.
* If negative, keep entire lower triangle.
* @param numUpper Number of subdiagonals to keep.
* If negative, keep entire upper triangle.
* @returns Rank `k` tensor of the same shape as input.
* The extracted banded tensor.
*/
/**
* @doc {heading:'Operations',
* subheading:'Linear Algebra',
* namespace:'linalg'}
*/
function bandPart_(a, numLower, numUpper) {
if (numLower % 1 !== 0) {
throw new Error("bandPart(): numLower must be an integer, got " + numLower + ".");
}
if (numUpper % 1 !== 0) {
throw new Error("bandPart(): numUpper must be an integer, got " + numUpper + ".");
}
var $a = tensor_util_env_1.convertToTensor(a, 'a', 'bandPart');
if ($a.rank < 2) {
throw new Error("bandPart(): Rank must be at least 2, got " + $a.rank + ".");
}
var shape = $a.shape, _a = $a.shape.slice(-2), M = _a[0], N = _a[1];
if (!(numLower <= M)) {
throw new Error("bandPart(): numLower (" + numLower + ")" +
(" must not be greater than the number of rows (" + M + ")."));
}
if (!(numUpper <= N)) {
throw new Error("bandPart(): numUpper (" + numUpper + ")" +
(" must not be greater than the number of columns (" + N + ")."));
}
if (numLower < 0) {
numLower = M;
}
if (numUpper < 0) {
numUpper = N;
}
var i = tensor_ops_1.range(0, M, 1, 'int32').reshape([-1, 1]), j = tensor_ops_1.range(0, N, 1, 'int32'), ij = binary_ops_1.sub(i, j);
var inBand = logical_ops_1.logicalAnd(ij.lessEqual(tensor_ops_1.scalar(+numLower, 'int32')), ij.greaterEqual(tensor_ops_1.scalar(-numUpper, 'int32')));
var zero = tensor_ops_1.zeros([M, N], $a.dtype);
return array_ops_1.stack(array_ops_1.unstack($a.reshape([-1, M, N])).map(function (mat) { return logical_ops_1.where(inBand, mat, zero); })).reshape(shape);
}
/**
* Gram-Schmidt orthogonalization.
*
* ```js
* const x = tf.tensor2d([[1, 2], [3, 4]]);
* let y = tf.linalg.gramSchmidt(x);
* y.print();
* console.log('Othogonalized:');
* y.dot(y.transpose()).print(); // should be nearly the identity matrix.
* console.log('First row direction maintained:');
* const data = await y.array();
* console.log(data[0][1] / data[0][0]); // should be nearly 2.
* ```
*
* @param xs The vectors to be orthogonalized, in one of the two following
* formats:
* - An Array of `tf.Tensor1D`.
* - A `tf.Tensor2D`, i.e., a matrix, in which case the vectors are the rows
* of `xs`.
* In each case, all the vectors must have the same length and the length
* must be greater than or equal to the number of vectors.
* @returns The orthogonalized and normalized vectors or matrix.
* Orthogonalization means that the vectors or the rows of the matrix
* are orthogonal (zero inner products). Normalization means that each
* vector or each row of the matrix has an L2 norm that equals `1`.
*/
/**
* @doc {heading:'Operations',
* subheading:'Linear Algebra',
* namespace:'linalg'}
*/
function gramSchmidt_(xs) {
var inputIsTensor2D;
if (Array.isArray(xs)) {
inputIsTensor2D = false;
util_1.assert(xs != null && xs.length > 0, function () { return 'Gram-Schmidt process: input must not be null, undefined, or ' +
'empty'; });
var dim_1 = xs[0].shape[0];
var _loop_1 = function (i) {
util_1.assert(xs[i].shape[0] === dim_1, function () {
return 'Gram-Schmidt: Non-unique lengths found in the input vectors: ' +
("(" + xs[i].shape[0] + " vs. " + dim_1 + ")");
});
};
for (var i = 1; i < xs.length; ++i) {
_loop_1(i);
}
}
else {
inputIsTensor2D = true;
xs = concat_split_1.split(xs, xs.shape[0], 0).map(function (x) { return array_ops_1.squeeze(x, [0]); });
}
util_1.assert(xs.length <= xs[0].shape[0], function () { return "Gram-Schmidt: Number of vectors (" + xs.length + ") exceeds " +
("number of dimensions (" + xs[0].shape[0] + ")."); });
var ys = [];
var xs1d = xs;
var _loop_2 = function (i) {
ys.push(engine_1.ENGINE.tidy(function () {
var x = xs1d[i];
if (i > 0) {
for (var j = 0; j < i; ++j) {
var proj = reduction_ops_1.sum(ys[j].mulStrict(x)).mul(ys[j]);
x = x.sub(proj);
}
}
return x.div(norm_1.norm(x, 'euclidean'));
}));
};
for (var i = 0; i < xs.length; ++i) {
_loop_2(i);
}
if (inputIsTensor2D) {
return array_ops_1.stack(ys, 0);
}
else {
return ys;
}
}
/**
* Compute QR decomposition of m-by-n matrix using Householder transformation.
*
* Implementation based on
* [http://www.cs.cornell.edu/~bindel/class/cs6210-f09/lec18.pdf]
* (http://www.cs.cornell.edu/~bindel/class/cs6210-f09/lec18.pdf)
*
* ```js
* const a = tf.tensor2d([[1, 2], [3, 4]]);
* let [q, r] = tf.linalg.qr(a);
* console.log('Q');
* q.print();
* console.log('R');
* r.print();
* console.log('Orthogonalized');
* q.dot(q.transpose()).print() // should be nearly the identity matrix.
* console.log('Reconstructed');
* q.dot(r).print(); // should be nearly [[1, 2], [3, 4]];
* ```
*
* @param x The `tf.Tensor` to be QR-decomposed. Must have rank >= 2. Suppose
* it has the shape `[..., M, N]`.
* @param fullMatrices An optional boolean parameter. Defaults to `false`.
* If `true`, compute full-sized `Q`. If `false` (the default),
* compute only the leading N columns of `Q` and `R`.
* @returns An `Array` of two `tf.Tensor`s: `[Q, R]`. `Q` is a unitary matrix,
* i.e., its columns all have unit norm and are mutually orthogonal.
* If `M >= N`,
* If `fullMatrices` is `false` (default),
* - `Q` has a shape of `[..., M, N]`,
* - `R` has a shape of `[..., N, N]`.
* If `fullMatrices` is `true` (default),
* - `Q` has a shape of `[..., M, M]`,
* - `R` has a shape of `[..., M, N]`.
* If `M < N`,
* - `Q` has a shape of `[..., M, M]`,
* - `R` has a shape of `[..., M, N]`.
* @throws If the rank of `x` is less than 2.
*/
/**
* @doc {heading:'Operations',
* subheading:'Linear Algebra',
* namespace:'linalg'}
*/
function qr_(x, fullMatrices) {
if (fullMatrices === void 0) { fullMatrices = false; }
if (x.rank < 2) {
throw new Error("qr() requires input tensor to have a rank >= 2, but got rank " + x.rank);
}
else if (x.rank === 2) {
return qr2d(x, fullMatrices);
}
else {
// Rank > 2.
// TODO(cais): Below we split the input into individual 2D tensors,
// perform QR decomposition on them and then stack the results back
// together. We should explore whether this can be parallelized.
var outerDimsProd = x.shape.slice(0, x.shape.length - 2)
.reduce(function (value, prev) { return value * prev; });
var x2ds = array_ops_1.unstack(x.reshape([
outerDimsProd, x.shape[x.shape.length - 2],
x.shape[x.shape.length - 1]
]), 0);
var q2ds_1 = [];
var r2ds_1 = [];
x2ds.forEach(function (x2d) {
var _a = qr2d(x2d, fullMatrices), q2d = _a[0], r2d = _a[1];
q2ds_1.push(q2d);
r2ds_1.push(r2d);
});
var q = array_ops_1.stack(q2ds_1, 0).reshape(x.shape);
var r = array_ops_1.stack(r2ds_1, 0).reshape(x.shape);
return [q, r];
}
}
function qr2d(x, fullMatrices) {
if (fullMatrices === void 0) { fullMatrices = false; }
return engine_1.ENGINE.tidy(function () {
if (x.shape.length !== 2) {
throw new Error("qr2d() requires a 2D Tensor, but got a " + x.shape.length + "D Tensor.");
}
var m = x.shape[0];
var n = x.shape[1];
var q = array_ops_1.eye(m); // Orthogonal transform so far.
var r = x.clone(); // Transformed matrix so far.
var one2D = tensor_ops_1.tensor2d([[1]], [1, 1]);
var w = one2D.clone();
var iters = m >= n ? n : m;
var _loop_3 = function (j) {
var _a;
// This tidy within the for-loop ensures we clean up temporary
// tensors as soon as they are no longer needed.
var rTemp = r;
var wTemp = w;
var qTemp = q;
_a = engine_1.ENGINE.tidy(function () {
// Find H = I - tau * w * w', to put zeros below R(j, j).
var rjEnd1 = r.slice([j, j], [m - j, 1]);
var normX = rjEnd1.norm();
var rjj = r.slice([j, j], [1, 1]);
// The sign() function returns 0 on 0, which causes division by zero.
var s = tensor_ops_1.tensor2d([[-1]]).where(rjj.greater(0), tensor_ops_1.tensor2d([[1]]));
var u1 = rjj.sub(s.mul(normX));
var wPre = rjEnd1.div(u1);
if (wPre.shape[0] === 1) {
w = one2D.clone();
}
else {
w = one2D.concat(wPre.slice([1, 0], [wPre.shape[0] - 1, wPre.shape[1]]), 0);
}
var tau = s.matMul(u1).div(normX).neg();
// -- R := HR, Q := QH.
var rjEndAll = r.slice([j, 0], [m - j, n]);
var tauTimesW = tau.mul(w);
if (j === 0) {
r = rjEndAll.sub(tauTimesW.matMul(w.transpose().matMul(rjEndAll)));
}
else {
var rTimesTau = rjEndAll.sub(tauTimesW.matMul(w.transpose().matMul(rjEndAll)));
r = r.slice([0, 0], [j, n]).concat(rTimesTau, 0);
}
var qAllJEnd = q.slice([0, j], [m, q.shape[1] - j]);
if (j === 0) {
q = qAllJEnd.sub(qAllJEnd.matMul(w).matMul(tauTimesW.transpose()));
}
else {
var qTimesTau = qAllJEnd.sub(qAllJEnd.matMul(w).matMul(tauTimesW.transpose()));
q = q.slice([0, 0], [m, j]).concat(qTimesTau, 1);
}
return [w, r, q];
}), w = _a[0], r = _a[1], q = _a[2];
globals_1.dispose([rTemp, wTemp, qTemp]);
};
for (var j = 0; j < iters; ++j) {
_loop_3(j);
}
if (!fullMatrices && m > n) {
q = q.slice([0, 0], [m, n]);
r = r.slice([0, 0], [n, n]);
}
return [q, r];
});
}
exports.bandPart = operation_1.op({ bandPart_: bandPart_ });
exports.gramSchmidt = operation_1.op({ gramSchmidt_: gramSchmidt_ });
exports.qr = operation_1.op({ qr_: qr_ });
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