/**
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* @license
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* Copyright 2018 Google LLC. All Rights Reserved.
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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* =============================================================================
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*/
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/**
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* Linear algebra ops.
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*/
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import {ENGINE} from '../engine';
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import {dispose} from '../globals';
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import {Tensor, Tensor1D, Tensor2D} from '../tensor';
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import {convertToTensor} from '../tensor_util_env';
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import {TensorLike} from '../types';
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import {assert} from '../util';
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import {eye, squeeze, stack, unstack} from './array_ops';
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import {sub} from './binary_ops';
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import {split} from './concat_split';
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import {logicalAnd, where} from './logical_ops';
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import {norm} from './norm';
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import {op} from './operation';
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import {sum} from './reduction_ops';
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import {range, scalar, tensor2d, zeros} from './tensor_ops';
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/**
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* Copy a tensor setting everything outside a central band in each innermost
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* matrix to zero.
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*
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* The band part is computed as follows: Assume input has `k` dimensions
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* `[I, J, K, ..., M, N]`, then the output is a tensor with the same shape where
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* `band[i, j, k, ..., m, n] = in_band(m, n) * input[i, j, k, ..., m, n]`.
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* The indicator function
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* `in_band(m, n) = (num_lower < 0 || (m-n) <= num_lower))`
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* `&& (num_upper < 0 || (n-m) <= num_upper)`
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*
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* ```js
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* const x = tf.tensor2d([[ 0, 1, 2, 3],
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* [-1, 0, 1, 2],
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* [-2, -1, 0, 1],
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* [-3, -2, -1, 0]]);
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* let y = tf.linalg.bandPart(x, 1, -1);
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* y.print(); // [[ 0, 1, 2, 3],
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* // [-1, 0, 1, 2],
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* // [ 0, -1, 0, 1],
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* // [ 0, 0 , -1, 0]]
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* let z = tf.linalg.bandPart(x, 2, 1);
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* z.print(); // [[ 0, 1, 0, 0],
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* // [-1, 0, 1, 0],
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* // [-2, -1, 0, 1],
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* // [ 0, -2, -1, 0]]
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* ```
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*
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* @param x Rank `k` tensor
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* @param numLower Number of subdiagonals to keep.
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* If negative, keep entire lower triangle.
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* @param numUpper Number of subdiagonals to keep.
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* If negative, keep entire upper triangle.
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* @returns Rank `k` tensor of the same shape as input.
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* The extracted banded tensor.
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*/
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/**
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* @doc {heading:'Operations',
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* subheading:'Linear Algebra',
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* namespace:'linalg'}
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*/
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function bandPart_<T extends Tensor>(
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a: T|TensorLike, numLower: number, numUpper: number
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): T
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{
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if( numLower%1 !== 0 ){
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throw new Error(
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`bandPart(): numLower must be an integer, got ${numLower}.`
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);
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}
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if( numUpper%1 !== 0 ){
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throw new Error(
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`bandPart(): numUpper must be an integer, got ${numUpper}.`
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);
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}
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const $a = convertToTensor(a,'a','bandPart');
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if( $a.rank < 2 ) {
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throw new Error(`bandPart(): Rank must be at least 2, got ${$a.rank}.`);
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}
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const shape = $a.shape,
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[M,N] = $a.shape.slice(-2);
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if( !(numLower <= M) ) {
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throw new Error(
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`bandPart(): numLower (${numLower})` +
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` must not be greater than the number of rows (${M}).`
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);
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}
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if( !(numUpper <= N) ) {
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throw new Error(
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`bandPart(): numUpper (${numUpper})` +
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` must not be greater than the number of columns (${N}).`
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);
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}
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if( numLower < 0 ) { numLower = M; }
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if( numUpper < 0 ) { numUpper = N; }
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const i = range(0,M, 1, 'int32').reshape([-1,1]),
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j = range(0,N, 1, 'int32'),
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ij = sub(i,j);
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const inBand = logicalAnd(
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ij. lessEqual( scalar(+numLower,'int32') ),
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ij.greaterEqual( scalar(-numUpper,'int32') )
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);
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const zero = zeros([M,N], $a.dtype);
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return stack(
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unstack( $a.reshape([-1,M,N]) ).map(
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mat => where(inBand, mat, zero)
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)
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).reshape(shape) as T;
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}
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/**
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* Gram-Schmidt orthogonalization.
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*
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* ```js
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* const x = tf.tensor2d([[1, 2], [3, 4]]);
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* let y = tf.linalg.gramSchmidt(x);
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* y.print();
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* console.log('Othogonalized:');
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* y.dot(y.transpose()).print(); // should be nearly the identity matrix.
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* console.log('First row direction maintained:');
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* const data = await y.array();
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* console.log(data[0][1] / data[0][0]); // should be nearly 2.
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* ```
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*
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* @param xs The vectors to be orthogonalized, in one of the two following
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* formats:
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* - An Array of `tf.Tensor1D`.
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* - A `tf.Tensor2D`, i.e., a matrix, in which case the vectors are the rows
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* of `xs`.
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* In each case, all the vectors must have the same length and the length
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* must be greater than or equal to the number of vectors.
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* @returns The orthogonalized and normalized vectors or matrix.
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* Orthogonalization means that the vectors or the rows of the matrix
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* are orthogonal (zero inner products). Normalization means that each
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* vector or each row of the matrix has an L2 norm that equals `1`.
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*/
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/**
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* @doc {heading:'Operations',
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* subheading:'Linear Algebra',
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* namespace:'linalg'}
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*/
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function gramSchmidt_(xs: Tensor1D[]|Tensor2D): Tensor1D[]|Tensor2D {
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let inputIsTensor2D: boolean;
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if (Array.isArray(xs)) {
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inputIsTensor2D = false;
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assert(
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xs != null && xs.length > 0,
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() => 'Gram-Schmidt process: input must not be null, undefined, or ' +
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'empty');
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const dim = xs[0].shape[0];
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for (let i = 1; i < xs.length; ++i) {
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assert(
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xs[i].shape[0] === dim,
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() =>
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'Gram-Schmidt: Non-unique lengths found in the input vectors: ' +
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`(${(xs as Tensor1D[])[i].shape[0]} vs. ${dim})`);
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}
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} else {
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inputIsTensor2D = true;
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xs = split(xs, xs.shape[0], 0).map(x => squeeze(x, [0]));
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}
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assert(
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xs.length <= xs[0].shape[0],
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() => `Gram-Schmidt: Number of vectors (${
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(xs as Tensor1D[]).length}) exceeds ` +
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`number of dimensions (${(xs as Tensor1D[])[0].shape[0]}).`);
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const ys: Tensor1D[] = [];
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const xs1d = xs;
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for (let i = 0; i < xs.length; ++i) {
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ys.push(ENGINE.tidy(() => {
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let x = xs1d[i];
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if (i > 0) {
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for (let j = 0; j < i; ++j) {
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const proj = sum(ys[j].mulStrict(x)).mul(ys[j]);
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x = x.sub(proj);
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}
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}
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return x.div(norm(x, 'euclidean'));
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}));
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}
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if (inputIsTensor2D) {
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return stack(ys, 0) as Tensor2D;
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} else {
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return ys;
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}
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}
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/**
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* Compute QR decomposition of m-by-n matrix using Householder transformation.
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*
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* Implementation based on
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* [http://www.cs.cornell.edu/~bindel/class/cs6210-f09/lec18.pdf]
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* (http://www.cs.cornell.edu/~bindel/class/cs6210-f09/lec18.pdf)
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*
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* ```js
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* const a = tf.tensor2d([[1, 2], [3, 4]]);
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* let [q, r] = tf.linalg.qr(a);
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* console.log('Q');
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* q.print();
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* console.log('R');
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* r.print();
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* console.log('Orthogonalized');
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* q.dot(q.transpose()).print() // should be nearly the identity matrix.
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* console.log('Reconstructed');
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* q.dot(r).print(); // should be nearly [[1, 2], [3, 4]];
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* ```
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*
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* @param x The `tf.Tensor` to be QR-decomposed. Must have rank >= 2. Suppose
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* it has the shape `[..., M, N]`.
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* @param fullMatrices An optional boolean parameter. Defaults to `false`.
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* If `true`, compute full-sized `Q`. If `false` (the default),
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* compute only the leading N columns of `Q` and `R`.
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* @returns An `Array` of two `tf.Tensor`s: `[Q, R]`. `Q` is a unitary matrix,
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* i.e., its columns all have unit norm and are mutually orthogonal.
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* If `M >= N`,
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* If `fullMatrices` is `false` (default),
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* - `Q` has a shape of `[..., M, N]`,
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* - `R` has a shape of `[..., N, N]`.
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* If `fullMatrices` is `true` (default),
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* - `Q` has a shape of `[..., M, M]`,
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* - `R` has a shape of `[..., M, N]`.
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* If `M < N`,
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* - `Q` has a shape of `[..., M, M]`,
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* - `R` has a shape of `[..., M, N]`.
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* @throws If the rank of `x` is less than 2.
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*/
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/**
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* @doc {heading:'Operations',
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* subheading:'Linear Algebra',
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* namespace:'linalg'}
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*/
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function qr_(x: Tensor, fullMatrices = false): [Tensor, Tensor] {
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if (x.rank < 2) {
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throw new Error(
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`qr() requires input tensor to have a rank >= 2, but got rank ${
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x.rank}`);
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} else if (x.rank === 2) {
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return qr2d(x as Tensor2D, fullMatrices);
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} else {
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// Rank > 2.
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// TODO(cais): Below we split the input into individual 2D tensors,
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// perform QR decomposition on them and then stack the results back
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// together. We should explore whether this can be parallelized.
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const outerDimsProd = x.shape.slice(0, x.shape.length - 2)
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.reduce((value, prev) => value * prev);
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const x2ds = unstack(
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x.reshape([
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outerDimsProd, x.shape[x.shape.length - 2],
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x.shape[x.shape.length - 1]
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]),
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0);
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const q2ds: Tensor2D[] = [];
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const r2ds: Tensor2D[] = [];
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x2ds.forEach(x2d => {
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const [q2d, r2d] = qr2d(x2d as Tensor2D, fullMatrices);
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q2ds.push(q2d);
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r2ds.push(r2d);
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});
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const q = stack(q2ds, 0).reshape(x.shape);
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const r = stack(r2ds, 0).reshape(x.shape);
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return [q, r];
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}
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}
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function qr2d(x: Tensor2D, fullMatrices = false): [Tensor2D, Tensor2D] {
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return ENGINE.tidy(() => {
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if (x.shape.length !== 2) {
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throw new Error(
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`qr2d() requires a 2D Tensor, but got a ${x.shape.length}D Tensor.`);
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}
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const m = x.shape[0];
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const n = x.shape[1];
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let q = eye(m); // Orthogonal transform so far.
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let r = x.clone(); // Transformed matrix so far.
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const one2D = tensor2d([[1]], [1, 1]);
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let w: Tensor2D = one2D.clone();
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const iters = m >= n ? n : m;
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for (let j = 0; j < iters; ++j) {
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// This tidy within the for-loop ensures we clean up temporary
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// tensors as soon as they are no longer needed.
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const rTemp = r;
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const wTemp = w;
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const qTemp = q;
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[w, r, q] = ENGINE.tidy((): [Tensor2D, Tensor2D, Tensor2D] => {
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// Find H = I - tau * w * w', to put zeros below R(j, j).
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const rjEnd1 = r.slice([j, j], [m - j, 1]);
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const normX = rjEnd1.norm();
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const rjj = r.slice([j, j], [1, 1]);
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// The sign() function returns 0 on 0, which causes division by zero.
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const s = tensor2d([[-1]]).where(rjj.greater(0), tensor2d([[1]]));
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const u1 = rjj.sub(s.mul(normX));
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const wPre = rjEnd1.div(u1);
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if (wPre.shape[0] === 1) {
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w = one2D.clone();
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} else {
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w = one2D.concat(
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wPre.slice([1, 0], [wPre.shape[0] - 1, wPre.shape[1]]) as
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Tensor2D,
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0);
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}
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const tau = s.matMul(u1).div(normX).neg() as Tensor2D;
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// -- R := HR, Q := QH.
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const rjEndAll = r.slice([j, 0], [m - j, n]);
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const tauTimesW: Tensor2D = tau.mul(w);
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if (j === 0) {
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r = rjEndAll.sub(tauTimesW.matMul(w.transpose().matMul(rjEndAll)));
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} else {
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const rTimesTau: Tensor2D =
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rjEndAll.sub(tauTimesW.matMul(w.transpose().matMul(rjEndAll)));
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r = r.slice([0, 0], [j, n]).concat(rTimesTau, 0);
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}
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const qAllJEnd = q.slice([0, j], [m, q.shape[1] - j]);
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if (j === 0) {
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q = qAllJEnd.sub(qAllJEnd.matMul(w).matMul(tauTimesW.transpose()));
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} else {
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const qTimesTau: Tensor2D =
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qAllJEnd.sub(qAllJEnd.matMul(w).matMul(tauTimesW.transpose()));
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q = q.slice([0, 0], [m, j]).concat(qTimesTau, 1);
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}
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return [w, r, q];
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});
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dispose([rTemp, wTemp, qTemp]);
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}
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if (!fullMatrices && m > n) {
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q = q.slice([0, 0], [m, n]);
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r = r.slice([0, 0], [n, n]);
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}
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return [q, r];
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}) as [Tensor2D, Tensor2D];
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}
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export const bandPart = op({bandPart_});
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export const gramSchmidt = op({gramSchmidt_});
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export const qr = op({qr_});
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