#!/usr/bin/env python # -*- coding: utf-8 -*- # Copyright 1999-2020 Alibaba Group Holding Ltd. # # 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. import itertools from collections.abc import Iterable import numpy as np from ... import opcodes as OperandDef from ...serialize import ValueType, KeyField, AnyField, TupleField, BoolField from ...utils import recursive_tile from ..utils import validate_axis from ..array_utils import device, as_same_device from ..operands import TensorHasInput, TensorOperandMixin from ..arithmetic import sqrt from ..datasource import tensor as astensor from .svd import svd class TensorNorm(TensorHasInput, TensorOperandMixin): _op_type_ = OperandDef.NORM _input = KeyField('input') _ord = AnyField('ord') _axis = TupleField('axis', ValueType.int32) _keepdims = BoolField('keepdims') def __init__(self, ord=None, axis=None, keepdims=None, dtype=None, sparse=False, **kw): super().__init__(_ord=ord, _axis=axis, _keepdims=keepdims, _dtype=dtype, _sparse=sparse, **kw) @property def ord(self): return getattr(self, '_ord', None) @property def axis(self): return self._axis @property def keepdims(self): return self._keepdims def _set_inputs(self, inputs): super()._set_inputs(inputs) self._input = self._inputs[0] def __call__(self, x): r = x.astype(self.dtype) shape = self._norm(r, self._ord, self._axis, self._keepdims).shape return self.new_tensor([x], shape) @classmethod def tile(cls, op): x = astensor(op.input) axis = op.axis ord = op.ord keepdims = op.keepdims axis_chunk_shapes = tuple(x.chunk_shape[i] for i in axis) can_apply_norm = all(s == 1 for s in axis_chunk_shapes) if can_apply_norm: axis_set = set(axis) get_shape = lambda shape: tuple(s if i not in axis_set else 1 for i, s in enumerate(shape) if i not in axis_set or keepdims) out_chunk_shape = get_shape(x.chunk_shape) out_chunks = [] for idx in itertools.product(*[range(s) for s in out_chunk_shape]): idx_iter = iter(idx) in_idx = tuple(0 if i in axis_set and not keepdims else next(idx_iter) for i in range(x.ndim)) c = x.cix[in_idx] chunk_op = op.copy().reset_key() out_chunk = chunk_op.new_chunk([c], shape=get_shape(c.shape), index=idx) out_chunks.append(out_chunk) nsplits = [tuple(c.shape[i] for c in out_chunks if all(idx == 0 for j, idx in enumerate(c.index) if j != i)) for i in range(len(out_chunks[0].shape))] new_op = op.copy() return new_op.new_tensors(op.inputs, op.outputs[0].shape, chunks=out_chunks, nsplits=nsplits) r = cls._norm(x.astype(op.outputs[0].dtype), ord, axis, keepdims) recursive_tile(r) new_op = op.copy() return new_op.new_tensors(op.inputs, op.outputs[0].shape, chunks=r.chunks, nsplits=r.nsplits) @staticmethod def _norm(r, ord, axis, keepdims): if ord is None: return sqrt((abs(r) ** 2).sum(axis=axis, keepdims=keepdims)) elif ord == 'nuc': if len(axis) == 1: raise ValueError('Invalid norm order for vectors.') return svd(r)[1][np.newaxis].sum(keepdims=keepdims) elif ord == np.inf: if r.ndim > 2: raise ValueError('Improper number of dimensions to norm.') r = abs(r) if len(axis) == 1: return r.max(axis=axis, keepdims=keepdims) else: return r.sum(axis=axis[1], keepdims=keepdims).max(keepdims=keepdims) elif ord == -np.inf: if r.ndim > 2: raise ValueError('Improper number of dimensions to norm.') r = abs(r) if len(axis) == 1: return r.min(axis=axis, keepdims=keepdims) else: return r.sum(axis=axis[1], keepdims=keepdims).min(keepdims=keepdims) elif ord == 0: if r.ndim > 2: raise ValueError('Improper number of dimensions to norm.') if len(axis) == 2: raise ValueError('Invalid norm order for matrices.') return (r != 0).astype(r.dtype).sum(axis=axis, keepdims=keepdims) elif ord == 1: if r.ndim > 2: raise ValueError('Improper number of dimensions to norm.') r = abs(r) if len(axis) == 1: return r.sum(axis=axis, keepdims=keepdims) else: return r.sum(axis=axis[0], keepdims=keepdims).max(keepdims=keepdims) elif ord == -1 and len(axis) == 2: if r.ndim > 2: raise ValueError('Improper number of dimensions to norm.') return abs(r).sum(axis=axis[0], keepdims=keepdims).min(keepdims=keepdims) elif ord == 2 and len(axis) == 2: return svd(r)[1][np.newaxis].max(keepdims=keepdims) elif ord == -2 and len(axis) == 2: return svd(r)[1][np.newaxis].min(keepdims=keepdims) else: if len(axis) == 2: raise ValueError('Invalid norm order for matrices.') return (abs(r) ** ord).sum(axis=axis, keepdims=keepdims) ** (1.0 / ord) @classmethod def execute(cls, ctx, op): (x,), device_id, xp = as_same_device( [ctx[c.key] for c in op.inputs], device=op.device, ret_extra=True) with device(device_id): ctx[op.outputs[0].key] = xp.linalg.norm(x, ord=op.ord, axis=op.axis, keepdims=op.keepdims) [docs]def norm(x, ord=None, axis=None, keepdims=False): r""" Matrix or vector norm. This function is able to return one of eight different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ``ord`` parameter. Parameters ---------- x : array_like Input tensor. If `axis` is None, `x` must be 1-D or 2-D. ord : {non-zero int, inf, -inf, 'fro', 'nuc'}, optional Order of the norm (see table under ``Notes``). inf means mars tensor's `inf` object. axis : {int, 2-tuple of ints, None}, optional If `axis` is an integer, it specifies the axis of `x` along which to compute the vector norms. If `axis` is a 2-tuple, it specifies the axes that hold 2-D matrices, and the matrix norms of these matrices are computed. If `axis` is None then either a vector norm (when `x` is 1-D) or a matrix norm (when `x` is 2-D) is returned. keepdims : bool, optional If this is set to True, the axes which are normed over are left in the result as dimensions with size one. With this option the result will broadcast correctly against the original `x`. Returns ------- n : float or Tensor Norm of the matrix or vector(s). Notes ----- For values of ``ord <= 0``, the result is, strictly speaking, not a mathematical 'norm', but it may still be useful for various numerical purposes. The following norms can be calculated: ===== ============================ ========================== ord norm for matrices norm for vectors ===== ============================ ========================== None Frobenius norm 2-norm 'fro' Frobenius norm -- 'nuc' nuclear norm -- inf max(sum(abs(x), axis=1)) max(abs(x)) -inf min(sum(abs(x), axis=1)) min(abs(x)) 0 -- sum(x != 0) 1 max(sum(abs(x), axis=0)) as below -1 min(sum(abs(x), axis=0)) as below 2 2-norm (largest sing. value) as below -2 smallest singular value as below other -- sum(abs(x)**ord)**(1./ord) ===== ============================ ========================== The Frobenius norm is given by [1]_: :math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}` The nuclear norm is the sum of the singular values. References ---------- .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15 Examples -------- >>> from mars.tensor import linalg as LA >>> import mars.tensor as mt >>> a = mt.arange(9) - 4 >>> a.execute() array([-4, -3, -2, -1, 0, 1, 2, 3, 4]) >>> b = a.reshape((3, 3)) >>> b.execute() array([[-4, -3, -2], [-1, 0, 1], [ 2, 3, 4]]) >>> LA.norm(a).execute() 7.745966692414834 >>> LA.norm(b).execute() 7.745966692414834 >>> LA.norm(b, 'fro').execute() 7.745966692414834 >>> LA.norm(a, mt.inf).execute() 4.0 >>> LA.norm(b, mt.inf).execute() 9.0 >>> LA.norm(a, -mt.inf).execute() 0.0 >>> LA.norm(b, -mt.inf).execute() 2.0 >>> LA.norm(a, 1).execute() 20.0 >>> LA.norm(b, 1).execute() 7.0 >>> LA.norm(a, -1).execute() 0.0 >>> LA.norm(b, -1).execute() 6.0 >>> LA.norm(a, 2).execute() 7.745966692414834 >>> LA.norm(b, 2).execute() 7.3484692283495345 >>> LA.norm(a, -2).execute() 0.0 >>> LA.norm(b, -2).execute() 4.351066026358965e-18 >>> LA.norm(a, 3).execute() 5.8480354764257312 >>> LA.norm(a, -3).execute() 0.0 Using the `axis` argument to compute vector norms: >>> c = mt.array([[ 1, 2, 3], ... [-1, 1, 4]]) >>> LA.norm(c, axis=0).execute() array([ 1.41421356, 2.23606798, 5. ]) >>> LA.norm(c, axis=1).execute() array([ 3.74165739, 4.24264069]) >>> LA.norm(c, ord=1, axis=1).execute() array([ 6., 6.]) Using the `axis` argument to compute matrix norms: >>> m = mt.arange(8).reshape(2,2,2) >>> LA.norm(m, axis=(1,2)).execute() array([ 3.74165739, 11.22497216]) >>> LA.norm(m[0, :, :]).execute(), LA.norm(m[1, :, :]).execute() (3.7416573867739413, 11.224972160321824) """ x = astensor(x) ndim = x.ndim if ord == 'fro': ord = None if axis is not None: if isinstance(axis, Iterable): axis = tuple(validate_axis(ndim, a) for a in axis) else: axis = (validate_axis(ndim, axis),) else: axis = tuple(range(x.ndim)) op = TensorNorm(ord=ord, axis=axis, keepdims=keepdims, dtype=np.result_type(x.dtype, np.float_), sparse=x.issparse()) return op(x)