from __future__ import annotations import operator import warnings from functools import partial from numbers import Number import numpy as np import tlz as toolz from dask.array.core import Array, concatenate, dotmany, from_delayed from dask.array.creation import eye from dask.array.random import RandomState, default_rng from dask.array.utils import ( array_safe, meta_from_array, solve_triangular_safe, svd_flip, ) from dask.base import tokenize, wait from dask.blockwise import blockwise from dask.delayed import delayed from dask.highlevelgraph import HighLevelGraph from dask.utils import apply, derived_from def _cumsum_blocks(it): total = 0 for x in it: total_previous = total total += x yield (total_previous, total) def _cumsum_part(last, new): return (last[1], last[1] + new) def _nanmin(m, n): k_0 = min([m, n]) k_1 = m if np.isnan(n) else n return k_1 if np.isnan(k_0) else k_0 def _wrapped_qr(a): """ A wrapper for np.linalg.qr that handles arrays with 0 rows Notes: Created for tsqr so as to manage cases with uncertain array dimensions. In particular, the case where arrays have (uncertain) chunks with 0 rows. """ # workaround may be removed when numpy stops rejecting edge cases if a.shape[0] == 0: return np.zeros_like(a, shape=(0, 0)), np.zeros_like(a, shape=(0, a.shape[1])) else: return np.linalg.qr(a) def tsqr(data, compute_svd=False, _max_vchunk_size=None): """Direct Tall-and-Skinny QR algorithm As presented in: A. Benson, D. Gleich, and J. Demmel. Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures. IEEE International Conference on Big Data, 2013. https://arxiv.org/abs/1301.1071 This algorithm is used to compute both the QR decomposition and the Singular Value Decomposition. It requires that the input array have a single column of blocks, each of which fit in memory. Parameters ---------- data: Array compute_svd: bool Whether to compute the SVD rather than the QR decomposition _max_vchunk_size: Integer Used internally in recursion to set the maximum row dimension of chunks in subsequent recursive calls. Notes ----- With ``k`` blocks of size ``(m, n)``, this algorithm has memory use that scales as ``k * n * n``. The implementation here is the recursive variant due to the ultimate need for one "single core" QR decomposition. In the non-recursive version of the algorithm, given ``k`` blocks, after ``k`` ``m * n`` QR decompositions, there will be a "single core" QR decomposition that will have to work with a ``(k * n, n)`` matrix. Here, recursion is applied as necessary to ensure that ``k * n`` is not larger than ``m`` (if ``m / n >= 2``). In particular, this is done to ensure that single core computations do not have to work on blocks larger than ``(m, n)``. Where blocks are irregular, the above logic is applied with the "height" of the "tallest" block used in place of ``m``. Consider use of the ``rechunk`` method to control this behavior. Taller blocks will reduce overall memory use (assuming that many of them still fit in memory at once). See Also -------- dask.array.linalg.qr Powered by this algorithm dask.array.linalg.svd Powered by this algorithm dask.array.linalg.sfqr Variant for short-and-fat arrays """ nr, nc = len(data.chunks[0]), len(data.chunks[1]) cr_max, cc = max(data.chunks[0]), data.chunks[1][0] if not (data.ndim == 2 and nc == 1): # Is a matrix # Only one column block raise ValueError( "Input must have the following properties:\n" " 1. Have two dimensions\n" " 2. Have only one column of blocks\n\n" "Note: This function (tsqr) supports QR decomposition in the case of\n" "tall-and-skinny matrices (single column chunk/block; see qr)\n" "Current shape: {},\nCurrent chunksize: {}".format( data.shape, data.chunksize ) ) token = "-" + tokenize(data, compute_svd) m, n = data.shape numblocks = (nr, 1) qq, rr = np.linalg.qr(np.ones(shape=(1, 1), dtype=data.dtype)) layers = data.__dask_graph__().layers.copy() dependencies = data.__dask_graph__().dependencies.copy() # Block qr name_qr_st1 = "qr" + token dsk_qr_st1 = blockwise( _wrapped_qr, name_qr_st1, "ij", data.name, "ij", numblocks={data.name: numblocks}, ) layers[name_qr_st1] = dsk_qr_st1 dependencies[name_qr_st1] = set(data.__dask_layers__()) # Block qr[0] name_q_st1 = "getitem" + token + "-q1" dsk_q_st1 = { (name_q_st1, i, 0): (operator.getitem, (name_qr_st1, i, 0), 0) for i in range(numblocks[0]) } layers[name_q_st1] = dsk_q_st1 dependencies[name_q_st1] = {name_qr_st1} # Block qr[1] name_r_st1 = "getitem" + token + "-r1" dsk_r_st1 = { (name_r_st1, i, 0): (operator.getitem, (name_qr_st1, i, 0), 1) for i in range(numblocks[0]) } layers[name_r_st1] = dsk_r_st1 dependencies[name_r_st1] = {name_qr_st1} # Next step is to obtain a QR decomposition for the stacked R factors, so either: # - gather R factors into a single core and do a QR decomposition # - recurse with tsqr (if single core computation too large and a-priori "meaningful # reduction" possible, meaning that chunks have to be well defined) single_core_compute_m = nr * cc chunks_well_defined = not any(np.isnan(c) for cs in data.chunks for c in cs) prospective_blocks = np.ceil(single_core_compute_m / cr_max) meaningful_reduction_possible = ( cr_max if _max_vchunk_size is None else _max_vchunk_size ) >= 2 * cc can_distribute = chunks_well_defined and int(prospective_blocks) > 1 if chunks_well_defined and meaningful_reduction_possible and can_distribute: # stack chunks into blocks and recurse using tsqr # Prepare to stack chunks into blocks (from block qr[1]) all_blocks = [] curr_block = [] curr_block_sz = 0 for idx, a_m in enumerate(data.chunks[0]): m_q = a_m n_q = min(m_q, cc) m_r = n_q # n_r = cc if curr_block_sz + m_r > cr_max: all_blocks.append(curr_block) curr_block = [] curr_block_sz = 0 curr_block.append((idx, m_r)) curr_block_sz += m_r if len(curr_block) > 0: all_blocks.append(curr_block) # R_stacked name_r_stacked = "stack" + token + "-r1" dsk_r_stacked = { (name_r_stacked, i, 0): ( np.vstack, (tuple, [(name_r_st1, idx, 0) for idx, _ in sub_block_info]), ) for i, sub_block_info in enumerate(all_blocks) } layers[name_r_stacked] = dsk_r_stacked dependencies[name_r_stacked] = {name_r_st1} # retrieve R_stacked for recursion with tsqr vchunks_rstacked = tuple( sum(map(lambda x: x[1], sub_block_info)) for sub_block_info in all_blocks ) graph = HighLevelGraph(layers, dependencies) # dsk.dependencies[name_r_stacked] = {data.name} r_stacked_meta = meta_from_array( data, len((sum(vchunks_rstacked), n)), dtype=rr.dtype ) r_stacked = Array( graph, name_r_stacked, shape=(sum(vchunks_rstacked), n), chunks=(vchunks_rstacked, n), meta=r_stacked_meta, ) # recurse q_inner, r_inner = tsqr(r_stacked, _max_vchunk_size=cr_max) layers = toolz.merge(q_inner.dask.layers, r_inner.dask.layers) dependencies = toolz.merge(q_inner.dask.dependencies, r_inner.dask.dependencies) # Q_inner: "unstack" name_q_st2 = "getitem" + token + "-q2" dsk_q_st2 = { (name_q_st2, j, 0): ( operator.getitem, (q_inner.name, i, 0), ((slice(e[0], e[1])), (slice(0, n))), ) for i, sub_block_info in enumerate(all_blocks) for j, e in zip( [x[0] for x in sub_block_info], _cumsum_blocks([x[1] for x in sub_block_info]), ) } layers[name_q_st2] = dsk_q_st2 dependencies[name_q_st2] = set(q_inner.__dask_layers__()) # R: R_inner name_r_st2 = "r-inner" + token dsk_r_st2 = {(name_r_st2, 0, 0): (r_inner.name, 0, 0)} layers[name_r_st2] = dsk_r_st2 dependencies[name_r_st2] = set(r_inner.__dask_layers__()) # Q: Block qr[0] (*) Q_inner name_q_st3 = "dot" + token + "-q3" dsk_q_st3 = blockwise( np.dot, name_q_st3, "ij", name_q_st1, "ij", name_q_st2, "ij", numblocks={name_q_st1: numblocks, name_q_st2: numblocks}, ) layers[name_q_st3] = dsk_q_st3 dependencies[name_q_st3] = {name_q_st1, name_q_st2} else: # Do single core computation # Stacking for in-core QR computation to_stack = [(name_r_st1, i, 0) for i in range(numblocks[0])] name_r_st1_stacked = "stack" + token + "-r1" dsk_r_st1_stacked = {(name_r_st1_stacked, 0, 0): (np.vstack, (tuple, to_stack))} layers[name_r_st1_stacked] = dsk_r_st1_stacked dependencies[name_r_st1_stacked] = {name_r_st1} # In-core QR computation name_qr_st2 = "qr" + token + "-qr2" dsk_qr_st2 = blockwise( np.linalg.qr, name_qr_st2, "ij", name_r_st1_stacked, "ij", numblocks={name_r_st1_stacked: (1, 1)}, ) layers[name_qr_st2] = dsk_qr_st2 dependencies[name_qr_st2] = {name_r_st1_stacked} # In-core qr[0] name_q_st2_aux = "getitem" + token + "-q2-aux" dsk_q_st2_aux = { (name_q_st2_aux, 0, 0): (operator.getitem, (name_qr_st2, 0, 0), 0) } layers[name_q_st2_aux] = dsk_q_st2_aux dependencies[name_q_st2_aux] = {name_qr_st2} chucks_are_all_known = not any(np.isnan(c) for cs in data.chunks for c in cs) if chucks_are_all_known: # when chunks are all known... # obtain slices on q from in-core compute (e.g.: (slice(10, 20), slice(0, 5))) q2_block_sizes = [min(e, n) for e in data.chunks[0]] block_slices = [ (slice(e[0], e[1]), slice(0, n)) for e in _cumsum_blocks(q2_block_sizes) ] dsk_q_blockslices = {} deps = set() else: # when chunks are not already known... # request shape information: vertical chunk sizes & column dimension (n) name_q2bs = "shape" + token + "-q2" dsk_q2_shapes = { (name_q2bs, i): (min, (getattr, (data.name, i, 0), "shape")) for i in range(numblocks[0]) } name_n = "getitem" + token + "-n" dsk_n = { name_n: (operator.getitem, (getattr, (data.name, 0, 0), "shape"), 1) } # cumulative sums (start, end) name_q2cs = "cumsum" + token + "-q2" dsk_q2_cumsum = {(name_q2cs, 0): [0, (name_q2bs, 0)]} for i in range(1, numblocks[0]): dsk_q2_cumsum[(name_q2cs, i)] = ( _cumsum_part, (name_q2cs, i - 1), (name_q2bs, i), ) # obtain slices on q from in-core compute (e.g.: (slice(10, 20), slice(0, 5))) name_blockslice = "slice" + token + "-q" dsk_block_slices = { (name_blockslice, i): ( tuple, [(apply, slice, (name_q2cs, i)), (slice, 0, name_n)], ) for i in range(numblocks[0]) } dsk_q_blockslices = toolz.merge( dsk_n, dsk_q2_shapes, dsk_q2_cumsum, dsk_block_slices ) deps = {data.name} block_slices = [(name_blockslice, i) for i in range(numblocks[0])] layers["q-blocksizes" + token] = dsk_q_blockslices dependencies["q-blocksizes" + token] = deps # In-core qr[0] unstacking name_q_st2 = "getitem" + token + "-q2" dsk_q_st2 = { (name_q_st2, i, 0): (operator.getitem, (name_q_st2_aux, 0, 0), b) for i, b in enumerate(block_slices) } layers[name_q_st2] = dsk_q_st2 if chucks_are_all_known: dependencies[name_q_st2] = {name_q_st2_aux} else: dependencies[name_q_st2] = {name_q_st2_aux, "q-blocksizes" + token} # Q: Block qr[0] (*) In-core qr[0] name_q_st3 = "dot" + token + "-q3" dsk_q_st3 = blockwise( np.dot, name_q_st3, "ij", name_q_st1, "ij", name_q_st2, "ij", numblocks={name_q_st1: numblocks, name_q_st2: numblocks}, ) layers[name_q_st3] = dsk_q_st3 dependencies[name_q_st3] = {name_q_st1, name_q_st2} # R: In-core qr[1] name_r_st2 = "getitem" + token + "-r2" dsk_r_st2 = {(name_r_st2, 0, 0): (operator.getitem, (name_qr_st2, 0, 0), 1)} layers[name_r_st2] = dsk_r_st2 dependencies[name_r_st2] = {name_qr_st2} if not compute_svd: is_unknown_m = np.isnan(data.shape[0]) or any( np.isnan(c) for c in data.chunks[0] ) is_unknown_n = np.isnan(data.shape[1]) or any( np.isnan(c) for c in data.chunks[1] ) if is_unknown_m and is_unknown_n: # assumption: m >= n q_shape = data.shape q_chunks = (data.chunks[0], (np.nan,)) r_shape = (np.nan, np.nan) r_chunks = ((np.nan,), (np.nan,)) elif is_unknown_m and not is_unknown_n: # assumption: m >= n q_shape = data.shape q_chunks = (data.chunks[0], (n,)) r_shape = (n, n) r_chunks = (n, n) elif not is_unknown_m and is_unknown_n: # assumption: m >= n q_shape = data.shape q_chunks = (data.chunks[0], (np.nan,)) r_shape = (np.nan, np.nan) r_chunks = ((np.nan,), (np.nan,)) else: q_shape = ( data.shape if data.shape[0] >= data.shape[1] else (data.shape[0], data.shape[0]) ) q_chunks = ( data.chunks if data.shape[0] >= data.shape[1] else (data.chunks[0], data.chunks[0]) ) r_shape = (n, n) if data.shape[0] >= data.shape[1] else data.shape r_chunks = r_shape # dsk.dependencies[name_q_st3] = {data.name} # dsk.dependencies[name_r_st2] = {data.name} graph = HighLevelGraph(layers, dependencies) q_meta = meta_from_array(data, len(q_shape), qq.dtype) r_meta = meta_from_array(data, len(r_shape), rr.dtype) q = Array(graph, name_q_st3, shape=q_shape, chunks=q_chunks, meta=q_meta) r = Array(graph, name_r_st2, shape=r_shape, chunks=r_chunks, meta=r_meta) return q, r else: # In-core SVD computation name_svd_st2 = "svd" + token + "-2" dsk_svd_st2 = blockwise( np.linalg.svd, name_svd_st2, "ij", name_r_st2, "ij", numblocks={name_r_st2: (1, 1)}, ) # svd[0] name_u_st2 = "getitem" + token + "-u2" dsk_u_st2 = {(name_u_st2, 0, 0): (operator.getitem, (name_svd_st2, 0, 0), 0)} # svd[1] name_s_st2 = "getitem" + token + "-s2" dsk_s_st2 = {(name_s_st2, 0): (operator.getitem, (name_svd_st2, 0, 0), 1)} # svd[2] name_v_st2 = "getitem" + token + "-v2" dsk_v_st2 = {(name_v_st2, 0, 0): (operator.getitem, (name_svd_st2, 0, 0), 2)} # Q * U name_u_st4 = "getitem" + token + "-u4" dsk_u_st4 = blockwise( dotmany, name_u_st4, "ij", name_q_st3, "ik", name_u_st2, "kj", numblocks={name_q_st3: numblocks, name_u_st2: (1, 1)}, ) layers[name_svd_st2] = dsk_svd_st2 dependencies[name_svd_st2] = {name_r_st2} layers[name_u_st2] = dsk_u_st2 dependencies[name_u_st2] = {name_svd_st2} layers[name_u_st4] = dsk_u_st4 dependencies[name_u_st4] = {name_q_st3, name_u_st2} layers[name_s_st2] = dsk_s_st2 dependencies[name_s_st2] = {name_svd_st2} layers[name_v_st2] = dsk_v_st2 dependencies[name_v_st2] = {name_svd_st2} uu, ss, vvh = np.linalg.svd(np.ones(shape=(1, 1), dtype=data.dtype)) k = _nanmin(m, n) # avoid RuntimeWarning with np.nanmin([m, n]) m_u = m n_u = int(k) if not np.isnan(k) else k n_s = n_u m_vh = n_u n_vh = n d_vh = max(m_vh, n_vh) # full matrix returned: but basically n graph = HighLevelGraph(layers, dependencies) u_meta = meta_from_array(data, len((m_u, n_u)), uu.dtype) s_meta = meta_from_array(data, len((n_s,)), ss.dtype) vh_meta = meta_from_array(data, len((d_vh, d_vh)), vvh.dtype) u = Array( graph, name_u_st4, shape=(m_u, n_u), chunks=(data.chunks[0], (n_u,)), meta=u_meta, ) s = Array(graph, name_s_st2, shape=(n_s,), chunks=((n_s,),), meta=s_meta) vh = Array( graph, name_v_st2, shape=(d_vh, d_vh), chunks=((n,), (n,)), meta=vh_meta ) return u, s, vh def sfqr(data, name=None): """Direct Short-and-Fat QR Currently, this is a quick hack for non-tall-and-skinny matrices which are one chunk tall and (unless they are one chunk wide) have chunks that are wider than they are tall Q [R_1 R_2 ...] = [A_1 A_2 ...] it computes the factorization Q R_1 = A_1, then computes the other R_k's in parallel. Parameters ---------- data: Array See Also -------- dask.array.linalg.qr Main user API that uses this function dask.array.linalg.tsqr Variant for tall-and-skinny case """ nr, nc = len(data.chunks[0]), len(data.chunks[1]) cr, cc = data.chunks[0][0], data.chunks[1][0] if not ( (data.ndim == 2) and (nr == 1) # Is a matrix and ( # Has exactly one block row (cr <= cc) or (nc == 1) # Chunking dimension on rows is at least that on cols or... ) ): # ... only one block col raise ValueError( "Input must have the following properties:\n" " 1. Have two dimensions\n" " 2. Have only one row of blocks\n" " 3. Either one column of blocks or (first) chunk size on cols\n" " is at most that on rows (e.g.: for a 5x20 matrix,\n" " chunks=((5), (8,4,8)) is fine, but chunks=((5), (4,8,8)) is not;\n" " still, prefer something simple like chunks=(5,10) or chunks=5)\n\n" "Note: This function (sfqr) supports QR decomposition in the case\n" "of short-and-fat matrices (single row chunk/block; see qr)" ) prefix = name or "sfqr-" + tokenize(data) prefix += "_" m, n = data.shape qq, rr = np.linalg.qr(np.ones(shape=(1, 1), dtype=data.dtype)) layers = data.__dask_graph__().layers.copy() dependencies = data.__dask_graph__().dependencies.copy() # data = A = [A_1 A_rest] name_A_1 = prefix + "A_1" name_A_rest = prefix + "A_rest" layers[name_A_1] = {(name_A_1, 0, 0): (data.name, 0, 0)} dependencies[name_A_1] = set(data.__dask_layers__()) layers[name_A_rest] = { (name_A_rest, 0, idx): (data.name, 0, 1 + idx) for idx in range(nc - 1) } if len(layers[name_A_rest]) > 0: dependencies[name_A_rest] = set(data.__dask_layers__()) else: dependencies[name_A_rest] = set() # Q R_1 = A_1 name_Q_R1 = prefix + "Q_R_1" name_Q = prefix + "Q" name_R_1 = prefix + "R_1" layers[name_Q_R1] = {(name_Q_R1, 0, 0): (np.linalg.qr, (name_A_1, 0, 0))} dependencies[name_Q_R1] = {name_A_1} layers[name_Q] = {(name_Q, 0, 0): (operator.getitem, (name_Q_R1, 0, 0), 0)} dependencies[name_Q] = {name_Q_R1} layers[name_R_1] = {(name_R_1, 0, 0): (operator.getitem, (name_Q_R1, 0, 0), 1)} dependencies[name_R_1] = {name_Q_R1} graph = HighLevelGraph(layers, dependencies) Q_meta = meta_from_array(data, len((m, min(m, n))), dtype=qq.dtype) R_1_meta = meta_from_array(data, len((min(m, n), cc)), dtype=rr.dtype) Q = Array(graph, name_Q, shape=(m, min(m, n)), chunks=(m, min(m, n)), meta=Q_meta) R_1 = Array(graph, name_R_1, shape=(min(m, n), cc), chunks=(cr, cc), meta=R_1_meta) # R = [R_1 Q'A_rest] Rs = [R_1] if nc > 1: A_rest_meta = meta_from_array(data, len((min(m, n), n - cc)), dtype=rr.dtype) A_rest = Array( graph, name_A_rest, shape=(min(m, n), n - cc), chunks=(cr, data.chunks[1][1:]), meta=A_rest_meta, ) Rs.append(Q.T.dot(A_rest)) R = concatenate(Rs, axis=1) return Q, R def compression_level(n, q, n_oversamples=10, min_subspace_size=20): """Compression level to use in svd_compressed Given the size ``n`` of a space, compress that that to one of size ``q`` plus n_oversamples. The oversampling allows for greater flexibility in finding an appropriate subspace, a low value is often enough (10 is already a very conservative choice, it can be further reduced). ``q + oversampling`` should not be larger than ``n``. In this specific implementation, ``q + n_oversamples`` is at least ``min_subspace_size``. Parameters ---------- n: int Column/row dimension of original matrix q: int Size of the desired subspace (the actual size will be bigger, because of oversampling, see ``da.linalg.compression_level``) n_oversamples: int, default=10 Number of oversamples used for generating the sampling matrix. min_subspace_size : int, default=20 Minimum subspace size. Examples -------- >>> compression_level(100, 10) 20 """ return min(max(min_subspace_size, q + n_oversamples), n) def compression_matrix( data, q, iterator="power", n_power_iter=0, n_oversamples=10, seed=None, compute=False, ): """Randomly sample matrix to find most active subspace This compression matrix returned by this algorithm can be used to compute both the QR decomposition and the Singular Value Decomposition. Parameters ---------- data: Array q: int Size of the desired subspace (the actual size will be bigger, because of oversampling, see ``da.linalg.compression_level``) iterator: {'power', 'QR'}, default='power' Define the technique used for iterations to cope with flat singular spectra or when the input matrix is very large. n_power_iter: int Number of power iterations, useful when the singular values decay slowly. Error decreases exponentially as `n_power_iter` increases. In practice, set `n_power_iter` <= 4. n_oversamples: int, default=10 Number of oversamples used for generating the sampling matrix. This value increases the size of the subspace computed, which is more accurate at the cost of efficiency. Results are rarely sensitive to this choice though and in practice a value of 10 is very commonly high enough. compute : bool Whether or not to compute data at each use. Recomputing the input while performing several passes reduces memory pressure, but means that we have to compute the input multiple times. This is a good choice if the data is larger than memory and cheap to recreate. References ---------- N. Halko, P. G. Martinsson, and J. A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev., Survey and Review section, Vol. 53, num. 2, pp. 217-288, June 2011 https://arxiv.org/abs/0909.4061 """ if iterator not in ["power", "QR"]: raise ValueError( f"Iterator '{iterator}' not valid, must one one of ['power', 'QR']" ) m, n = data.shape comp_level = compression_level(min(m, n), q, n_oversamples=n_oversamples) if isinstance(seed, RandomState): state = seed else: state = default_rng(seed) datatype = np.float64 if (data.dtype).type in {np.float32, np.complex64}: datatype = np.float32 omega = state.standard_normal( size=(n, comp_level), chunks=(data.chunks[1], (comp_level,)) ).astype(datatype, copy=False) mat_h = data.dot(omega) if iterator == "power": for _ in range(n_power_iter): if compute: mat_h = mat_h.persist() wait(mat_h) tmp = data.T.dot(mat_h) if compute: tmp = tmp.persist() wait(tmp) mat_h = data.dot(tmp) q, _ = tsqr(mat_h) else: q, _ = tsqr(mat_h) for _ in range(n_power_iter): if compute: q = q.persist() wait(q) q, _ = tsqr(data.T.dot(q)) if compute: q = q.persist() wait(q) q, _ = tsqr(data.dot(q)) return q.T def svd_compressed( a, k, iterator="power", n_power_iter=0, n_oversamples=10, seed=None, compute=False, coerce_signs=True, ): """Randomly compressed rank-k thin Singular Value Decomposition. This computes the approximate singular value decomposition of a large array. This algorithm is generally faster than the normal algorithm but does not provide exact results. One can balance between performance and accuracy with input parameters (see below). Parameters ---------- a: Array Input array k: int Rank of the desired thin SVD decomposition. iterator: {'power', 'QR'}, default='power' Define the technique used for iterations to cope with flat singular spectra or when the input matrix is very large. n_power_iter: int, default=0 Number of power iterations, useful when the singular values decay slowly. Error decreases exponentially as `n_power_iter` increases. In practice, set `n_power_iter` <= 4. n_oversamples: int, default=10 Number of oversamples used for generating the sampling matrix. This value increases the size of the subspace computed, which is more accurate at the cost of efficiency. Results are rarely sensitive to this choice though and in practice a value of 10 is very commonly high enough. compute : bool Whether or not to compute data at each use. Recomputing the input while performing several passes reduces memory pressure, but means that we have to compute the input multiple times. This is a good choice if the data is larger than memory and cheap to recreate. coerce_signs : bool Whether or not to apply sign coercion to singular vectors in order to maintain deterministic results, by default True. Examples -------- >>> u, s, v = svd_compressed(x, 20) # doctest: +SKIP Returns ------- u: Array, unitary / orthogonal s: Array, singular values in decreasing order (largest first) v: Array, unitary / orthogonal References ---------- N. Halko, P. G. Martinsson, and J. A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev., Survey and Review section, Vol. 53, num. 2, pp. 217-288, June 2011 https://arxiv.org/abs/0909.4061 """ comp = compression_matrix( a, k, iterator=iterator, n_power_iter=n_power_iter, n_oversamples=n_oversamples, seed=seed, compute=compute, ) if compute: comp = comp.persist() wait(comp) a_compressed = comp.dot(a) v, s, u = tsqr(a_compressed.T, compute_svd=True) u = comp.T.dot(u.T) v = v.T u = u[:, :k] s = s[:k] v = v[:k, :] if coerce_signs: u, v = svd_flip(u, v) return u, s, v def qr(a): """ Compute the qr factorization of a matrix. Parameters ---------- a : Array Returns ------- q: Array, orthonormal r: Array, upper-triangular Examples -------- >>> q, r = da.linalg.qr(x) # doctest: +SKIP See Also -------- numpy.linalg.qr: Equivalent NumPy Operation dask.array.linalg.tsqr: Implementation for tall-and-skinny arrays dask.array.linalg.sfqr: Implementation for short-and-fat arrays """ if len(a.chunks[1]) == 1 and len(a.chunks[0]) > 1: return tsqr(a) elif len(a.chunks[0]) == 1: return sfqr(a) else: raise NotImplementedError( "qr currently supports only tall-and-skinny (single column chunk/block; see tsqr)\n" "and short-and-fat (single row chunk/block; see sfqr) matrices\n\n" "Consider use of the rechunk method. For example,\n\n" "x.rechunk({0: -1, 1: 'auto'}) or x.rechunk({0: 'auto', 1: -1})\n\n" "which rechunk one shorter axis to a single chunk, while allowing\n" "the other axis to automatically grow/shrink appropriately." ) def svd(a, coerce_signs=True): """ Compute the singular value decomposition of a matrix. Parameters ---------- a : (M, N) Array coerce_signs : bool Whether or not to apply sign coercion to singular vectors in order to maintain deterministic results, by default True. Examples -------- >>> u, s, v = da.linalg.svd(x) # doctest: +SKIP Returns ------- u : (M, K) Array, unitary / orthogonal Left-singular vectors of `a` (in columns) with shape (M, K) where K = min(M, N). s : (K,) Array, singular values in decreasing order (largest first) Singular values of `a`. v : (K, N) Array, unitary / orthogonal Right-singular vectors of `a` (in rows) with shape (K, N) where K = min(M, N). Warnings -------- SVD is only supported for arrays with chunking in one dimension. This requires that all inputs either contain a single column of chunks (tall-and-skinny) or a single row of chunks (short-and-fat). For arrays with chunking in both dimensions, see da.linalg.svd_compressed. See Also -------- np.linalg.svd : Equivalent NumPy Operation da.linalg.svd_compressed : Randomized SVD for fully chunked arrays dask.array.linalg.tsqr : QR factorization for tall-and-skinny arrays dask.array.utils.svd_flip : Sign normalization for singular vectors """ nb = a.numblocks if a.ndim != 2: raise ValueError( "Array must be 2D.\n" "Input shape: {}\n" "Input ndim: {}\n".format(a.shape, a.ndim) ) if nb[0] > 1 and nb[1] > 1: raise NotImplementedError( "Array must be chunked in one dimension only. " "This function (svd) only supports tall-and-skinny or short-and-fat " "matrices (see da.linalg.svd_compressed for SVD on fully chunked arrays).\n" "Input shape: {}\n" "Input numblocks: {}\n".format(a.shape, nb) ) # Single-chunk case if nb[0] == nb[1] == 1: m, n = a.shape k = min(a.shape) mu, ms, mv = np.linalg.svd( np.ones_like(a._meta, shape=(1, 1), dtype=a._meta.dtype) ) u, s, v = delayed(np.linalg.svd, nout=3)(a, full_matrices=False) u = from_delayed(u, shape=(m, k), meta=mu) s = from_delayed(s, shape=(k,), meta=ms) v = from_delayed(v, shape=(k, n), meta=mv) # Multi-chunk cases else: # Tall-and-skinny case if nb[0] > nb[1]: u, s, v = tsqr(a, compute_svd=True) truncate = a.shape[0] < a.shape[1] # Short-and-fat case else: vt, s, ut = tsqr(a.T, compute_svd=True) u, s, v = ut.T, s, vt.T truncate = a.shape[0] > a.shape[1] # Only when necessary, remove extra singular vectors if array # has shape that contradicts chunking, e.g. the array is a # column of chunks but still has more columns than rows overall if truncate: k = min(a.shape) u, v = u[:, :k], v[:k, :] if coerce_signs: u, v = svd_flip(u, v) return u, s, v def _solve_triangular_lower(a, b): return solve_triangular_safe(a, b, lower=True) def lu(a): """ Compute the lu decomposition of a matrix. Examples -------- >>> p, l, u = da.linalg.lu(x) # doctest: +SKIP Returns ------- p: Array, permutation matrix l: Array, lower triangular matrix with unit diagonal. u: Array, upper triangular matrix """ import scipy.linalg if a.ndim != 2: raise ValueError("Dimension must be 2 to perform lu decomposition") xdim, ydim = a.shape if xdim != ydim: raise ValueError("Input must be a square matrix to perform lu decomposition") if not len(set(a.chunks[0] + a.chunks[1])) == 1: msg = ( "All chunks must be a square matrix to perform lu decomposition. " "Use .rechunk method to change the size of chunks." ) raise ValueError(msg) vdim = len(a.chunks[0]) hdim = len(a.chunks[1]) token = tokenize(a) name_lu = "lu-lu-" + token name_p = "lu-p-" + token name_l = "lu-l-" + token name_u = "lu-u-" + token # for internal calculation name_p_inv = "lu-p-inv-" + token name_l_permuted = "lu-l-permute-" + token name_u_transposed = "lu-u-transpose-" + token name_plu_dot = "lu-plu-dot-" + token name_lu_dot = "lu-lu-dot-" + token dsk = {} for i in range(min(vdim, hdim)): target = (a.name, i, i) if i > 0: prevs = [] for p in range(i): prev = name_plu_dot, i, p, p, i dsk[prev] = (np.dot, (name_l_permuted, i, p), (name_u, p, i)) prevs.append(prev) target = (operator.sub, target, (sum, prevs)) # diagonal block dsk[name_lu, i, i] = (scipy.linalg.lu, target) # sweep to horizontal for j in range(i + 1, hdim): target = (np.dot, (name_p_inv, i, i), (a.name, i, j)) if i > 0: prevs = [] for p in range(i): prev = name_lu_dot, i, p, p, j dsk[prev] = (np.dot, (name_l, i, p), (name_u, p, j)) prevs.append(prev) target = (operator.sub, target, (sum, prevs)) dsk[name_lu, i, j] = (_solve_triangular_lower, (name_l, i, i), target) # sweep to vertical for k in range(i + 1, vdim): target = (a.name, k, i) if i > 0: prevs = [] for p in range(i): prev = name_plu_dot, k, p, p, i dsk[prev] = (np.dot, (name_l_permuted, k, p), (name_u, p, i)) prevs.append(prev) target = (operator.sub, target, (sum, prevs)) # solving x.dot(u) = target is equal to u.T.dot(x.T) = target.T dsk[name_lu, k, i] = ( np.transpose, ( _solve_triangular_lower, (name_u_transposed, i, i), (np.transpose, target), ), ) for i in range(min(vdim, hdim)): for j in range(min(vdim, hdim)): if i == j: dsk[name_p, i, j] = (operator.getitem, (name_lu, i, j), 0) dsk[name_l, i, j] = (operator.getitem, (name_lu, i, j), 1) dsk[name_u, i, j] = (operator.getitem, (name_lu, i, j), 2) # permuted l is required to be propagated to i > j blocks dsk[name_l_permuted, i, j] = (np.dot, (name_p, i, j), (name_l, i, j)) dsk[name_u_transposed, i, j] = (np.transpose, (name_u, i, j)) # transposed permutation matrix is equal to its inverse dsk[name_p_inv, i, j] = (np.transpose, (name_p, i, j)) elif i > j: dsk[name_p, i, j] = (np.zeros, (a.chunks[0][i], a.chunks[1][j])) # calculations are performed using permuted l, # thus the result should be reverted by inverted (=transposed) p # to have the same row order as diagonal blocks dsk[name_l, i, j] = (np.dot, (name_p_inv, i, i), (name_lu, i, j)) dsk[name_u, i, j] = (np.zeros, (a.chunks[0][i], a.chunks[1][j])) dsk[name_l_permuted, i, j] = (name_lu, i, j) else: dsk[name_p, i, j] = (np.zeros, (a.chunks[0][i], a.chunks[1][j])) dsk[name_l, i, j] = (np.zeros, (a.chunks[0][i], a.chunks[1][j])) dsk[name_u, i, j] = (name_lu, i, j) # l_permuted is not referred in upper triangulars pp, ll, uu = scipy.linalg.lu(np.ones(shape=(1, 1), dtype=a.dtype)) pp_meta = meta_from_array(a, dtype=pp.dtype) ll_meta = meta_from_array(a, dtype=ll.dtype) uu_meta = meta_from_array(a, dtype=uu.dtype) graph = HighLevelGraph.from_collections(name_p, dsk, dependencies=[a]) p = Array(graph, name_p, shape=a.shape, chunks=a.chunks, meta=pp_meta) graph = HighLevelGraph.from_collections(name_l, dsk, dependencies=[a]) l = Array(graph, name_l, shape=a.shape, chunks=a.chunks, meta=ll_meta) graph = HighLevelGraph.from_collections(name_u, dsk, dependencies=[a]) u = Array(graph, name_u, shape=a.shape, chunks=a.chunks, meta=uu_meta) return p, l, u def solve_triangular(a, b, lower=False): """ Solve the equation `a x = b` for `x`, assuming a is a triangular matrix. Parameters ---------- a : (M, M) array_like A triangular matrix b : (M,) or (M, N) array_like Right-hand side matrix in `a x = b` lower : bool, optional Use only data contained in the lower triangle of `a`. Default is to use upper triangle. Returns ------- x : (M,) or (M, N) array Solution to the system `a x = b`. Shape of return matches `b`. """ if a.ndim != 2: raise ValueError("a must be 2 dimensional") if b.ndim <= 2: if a.shape[1] != b.shape[0]: raise ValueError("a.shape[1] and b.shape[0] must be equal") if a.chunks[1] != b.chunks[0]: msg = ( "a.chunks[1] and b.chunks[0] must be equal. " "Use .rechunk method to change the size of chunks." ) raise ValueError(msg) else: raise ValueError("b must be 1 or 2 dimensional") vchunks = len(a.chunks[1]) hchunks = 1 if b.ndim == 1 else len(b.chunks[1]) token = tokenize(a, b, lower) name = "solve-triangular-" + token # for internal calculation # (name, i, j, k, l) corresponds to a_ij.dot(b_kl) name_mdot = "solve-tri-dot-" + token def _b_init(i, j): if b.ndim == 1: return b.name, i else: return b.name, i, j def _key(i, j): if b.ndim == 1: return name, i else: return name, i, j dsk = {} if lower: for i in range(vchunks): for j in range(hchunks): target = _b_init(i, j) if i > 0: prevs = [] for k in range(i): prev = name_mdot, i, k, k, j dsk[prev] = (np.dot, (a.name, i, k), _key(k, j)) prevs.append(prev) target = (operator.sub, target, (sum, prevs)) dsk[_key(i, j)] = (_solve_triangular_lower, (a.name, i, i), target) else: for i in range(vchunks): for j in range(hchunks): target = _b_init(i, j) if i < vchunks - 1: prevs = [] for k in range(i + 1, vchunks): prev = name_mdot, i, k, k, j dsk[prev] = (np.dot, (a.name, i, k), _key(k, j)) prevs.append(prev) target = (operator.sub, target, (sum, prevs)) dsk[_key(i, j)] = ( solve_triangular_safe, (a.name, i, i), target, ) graph = HighLevelGraph.from_collections(name, dsk, dependencies=[a, b]) a_meta = meta_from_array(a) b_meta = meta_from_array(b) res = _solve_triangular_lower( array_safe([[1, 0], [1, 2]], dtype=a.dtype, like=a_meta), array_safe([0, 1], dtype=b.dtype, like=b_meta), ) meta = meta_from_array(a, b.ndim, dtype=res.dtype) return Array(graph, name, shape=b.shape, chunks=b.chunks, meta=meta) def solve(a, b, sym_pos=None, assume_a="gen"): """ Solve the equation ``a x = b`` for ``x``. By default, use LU decomposition and forward / backward substitutions. When ``assume_a = "pos"`` use Cholesky decomposition. Parameters ---------- a : (M, M) array_like A square matrix. b : (M,) or (M, N) array_like Right-hand side matrix in ``a x = b``. sym_pos : bool, optional Assume a is symmetric and positive definite. If ``True``, use Cholesky decomposition. .. note:: ``sym_pos`` is deprecated and will be removed in a future version. Use ``assume_a = 'pos'`` instead. assume_a : {'gen', 'pos'}, optional Type of data matrix. It is used to choose the dedicated solver. Note that Dask does not support 'her' and 'sym' types. .. versionchanged:: 2022.8.0 ``assume_a = 'pos'`` was previously defined as ``sym_pos = True``. Returns ------- x : (M,) or (M, N) Array Solution to the system ``a x = b``. Shape of the return matches the shape of `b`. See Also -------- scipy.linalg.solve """ if sym_pos is not None: warnings.warn( "The sym_pos keyword is deprecated and should be replaced by using ``assume_a = 'pos'``." "``sym_pos`` will be removed in a future version.", category=FutureWarning, ) if sym_pos: assume_a = "pos" if assume_a == "pos": l, u = _cholesky(a) elif assume_a == "gen": p, l, u = lu(a) b = p.T.dot(b) else: raise ValueError( f"{assume_a = } is not a recognized matrix structure, " # noqa: E251 "valid structures in Dask are 'pos' and 'gen'." ) uy = solve_triangular(l, b, lower=True) return solve_triangular(u, uy) def inv(a): """ Compute the inverse of a matrix with LU decomposition and forward / backward substitutions. Parameters ---------- a : array_like Square matrix to be inverted. Returns ------- ainv : Array Inverse of the matrix `a`. """ return solve(a, eye(a.shape[0], chunks=a.chunks[0][0])) def _cholesky_lower(a): return np.linalg.cholesky(a) def cholesky(a, lower=False): """ Returns the Cholesky decomposition, :math:`A = L L^*` or :math:`A = U^* U` of a Hermitian positive-definite matrix A. Parameters ---------- a : (M, M) array_like Matrix to be decomposed lower : bool, optional Whether to compute the upper or lower triangular Cholesky factorization. Default is upper-triangular. Returns ------- c : (M, M) Array Upper- or lower-triangular Cholesky factor of `a`. """ l, u = _cholesky(a) if lower: return l else: return u def _cholesky(a): """ Private function to perform Cholesky decomposition, which returns both lower and upper triangulars. """ if a.ndim != 2: raise ValueError("Dimension must be 2 to perform cholesky decomposition") xdim, ydim = a.shape if xdim != ydim: raise ValueError( "Input must be a square matrix to perform cholesky decomposition" ) if not len(set(a.chunks[0] + a.chunks[1])) == 1: msg = ( "All chunks must be a square matrix to perform cholesky decomposition. " "Use .rechunk method to change the size of chunks." ) raise ValueError(msg) vdim = len(a.chunks[0]) hdim = len(a.chunks[1]) token = tokenize(a) name = "cholesky-" + token # (name_lt_dot, i, j, k, l) corresponds to l_ij.dot(l_kl.T) name_lt_dot = "cholesky-lt-dot-" + token # because transposed results are needed for calculation, # we can build graph for upper triangular simultaneously name_upper = "cholesky-upper-" + token # calculates lower triangulars because subscriptions get simpler dsk = {} for i in range(vdim): for j in range(hdim): if i < j: dsk[name, i, j] = ( partial(np.zeros_like, shape=(a.chunks[0][i], a.chunks[1][j])), meta_from_array(a), ) dsk[name_upper, j, i] = (name, i, j) elif i == j: target = (a.name, i, j) if i > 0: prevs = [] for p in range(i): prev = name_lt_dot, i, p, i, p dsk[prev] = (np.dot, (name, i, p), (name_upper, p, i)) prevs.append(prev) target = (operator.sub, target, (sum, prevs)) dsk[name, i, i] = (_cholesky_lower, target) dsk[name_upper, i, i] = (np.transpose, (name, i, i)) else: # solving x.dot(L11.T) = (A21 - L20.dot(L10.T)) is equal to # L11.dot(x.T) = A21.T - L10.dot(L20.T) # L11.dot(x.T) = A12 - L10.dot(L02) target = (a.name, j, i) if j > 0: prevs = [] for p in range(j): prev = name_lt_dot, j, p, i, p dsk[prev] = (np.dot, (name, j, p), (name_upper, p, i)) prevs.append(prev) target = (operator.sub, target, (sum, prevs)) dsk[name_upper, j, i] = (_solve_triangular_lower, (name, j, j), target) dsk[name, i, j] = (np.transpose, (name_upper, j, i)) graph_upper = HighLevelGraph.from_collections(name_upper, dsk, dependencies=[a]) graph_lower = HighLevelGraph.from_collections(name, dsk, dependencies=[a]) a_meta = meta_from_array(a) cho = np.linalg.cholesky(array_safe([[1, 2], [2, 5]], dtype=a.dtype, like=a_meta)) meta = meta_from_array(a, dtype=cho.dtype) lower = Array(graph_lower, name, shape=a.shape, chunks=a.chunks, meta=meta) # do not use .T, because part of transposed blocks are already calculated upper = Array(graph_upper, name_upper, shape=a.shape, chunks=a.chunks, meta=meta) return lower, upper def _reverse(x): return x[::-1] def lstsq(a, b): """ Return the least-squares solution to a linear matrix equation using QR decomposition. Solves the equation `a x = b` by computing a vector `x` that minimizes the Euclidean 2-norm `|| b - a x ||^2`. The equation may be under-, well-, or over- determined (i.e., the number of linearly independent rows of `a` can be less than, equal to, or greater than its number of linearly independent columns). If `a` is square and of full rank, then `x` (but for round-off error) is the "exact" solution of the equation. Parameters ---------- a : (M, N) array_like "Coefficient" matrix. b : {(M,), (M, K)} array_like Ordinate or "dependent variable" values. If `b` is two-dimensional, the least-squares solution is calculated for each of the `K` columns of `b`. Returns ------- x : {(N,), (N, K)} Array Least-squares solution. If `b` is two-dimensional, the solutions are in the `K` columns of `x`. residuals : {(1,), (K,)} Array Sums of residuals; squared Euclidean 2-norm for each column in ``b - a*x``. If `b` is 1-dimensional, this is a (1,) shape array. Otherwise the shape is (K,). rank : Array Rank of matrix `a`. s : (min(M, N),) Array Singular values of `a`. """ q, r = qr(a) x = solve_triangular(r, q.T.conj().dot(b)) residuals = b - a.dot(x) residuals = abs(residuals**2).sum(axis=0, keepdims=b.ndim == 1) token = tokenize(a, b) # r must be a triangular with single block # rank rname = "lstsq-rank-" + token rdsk = {(rname,): (np.linalg.matrix_rank, (r.name, 0, 0))} graph = HighLevelGraph.from_collections(rname, rdsk, dependencies=[r]) # rank must be an integer rank = Array(graph, rname, shape=(), chunks=(), dtype=int) # singular sname = "lstsq-singular-" + token rt = r.T.conj() sdsk = { (sname, 0): ( _reverse, (np.sqrt, (np.linalg.eigvalsh, (np.dot, (rt.name, 0, 0), (r.name, 0, 0)))), ) } graph = HighLevelGraph.from_collections(sname, sdsk, dependencies=[rt, r]) meta = meta_from_array(residuals, 1) s = Array(graph, sname, shape=(r.shape[0],), chunks=r.shape[0], meta=meta) return x, residuals, rank, s @derived_from(np.linalg) def norm(x, ord=None, axis=None, keepdims=False): if axis is None: axis = tuple(range(x.ndim)) elif isinstance(axis, Number): axis = (int(axis),) else: axis = tuple(axis) if len(axis) > 2: raise ValueError("Improper number of dimensions to norm.") if ord == "fro": ord = None if len(axis) == 1: raise ValueError("Invalid norm order for vectors.") r = abs(x) if ord is None: r = (r**2).sum(axis=axis, keepdims=keepdims) ** 0.5 elif ord == "nuc": if len(axis) == 1: raise ValueError("Invalid norm order for vectors.") if x.ndim > 2: raise NotImplementedError("SVD based norm not implemented for ndim > 2") r = svd(x)[1][None].sum(keepdims=keepdims) elif ord == np.inf: if len(axis) == 1: r = r.max(axis=axis, keepdims=keepdims) else: r = r.sum(axis=axis[1], keepdims=True).max(axis=axis[0], keepdims=True) if keepdims is False: r = r.squeeze(axis=axis) elif ord == -np.inf: if len(axis) == 1: r = r.min(axis=axis, keepdims=keepdims) else: r = r.sum(axis=axis[1], keepdims=True).min(axis=axis[0], keepdims=True) if keepdims is False: r = r.squeeze(axis=axis) elif ord == 0: if len(axis) == 2: raise ValueError("Invalid norm order for matrices.") r = (r != 0).astype(r.dtype).sum(axis=axis, keepdims=keepdims) elif ord == 1: if len(axis) == 1: r = r.sum(axis=axis, keepdims=keepdims) else: r = r.sum(axis=axis[0], keepdims=True).max(axis=axis[1], keepdims=True) if keepdims is False: r = r.squeeze(axis=axis) elif len(axis) == 2 and ord == -1: r = r.sum(axis=axis[0], keepdims=True).min(axis=axis[1], keepdims=True) if keepdims is False: r = r.squeeze(axis=axis) elif len(axis) == 2 and ord == 2: if x.ndim > 2: raise NotImplementedError("SVD based norm not implemented for ndim > 2") r = svd(x)[1][None].max(keepdims=keepdims) elif len(axis) == 2 and ord == -2: if x.ndim > 2: raise NotImplementedError("SVD based norm not implemented for ndim > 2") r = svd(x)[1][None].min(keepdims=keepdims) else: if len(axis) == 2: raise ValueError("Invalid norm order for matrices.") r = (r**ord).sum(axis=axis, keepdims=keepdims) ** (1.0 / ord) return r