lsimodel.py

#!/usr/bin/env python
# -*- coding: utf-8 -*-
#
# Copyright (C) 2010 Radim Rehurek <radimrehurek@seznam.cz>
# Licensed under the GNU LGPL v2.1 - https://www.gnu.org/licenses/old-licenses/lgpl-2.1.en.html


"""Module for `Latent Semantic Analysis (aka Latent Semantic Indexing)
<https://en.wikipedia.org/wiki/Latent_semantic_analysis#Latent_semantic_indexing>`_.

Implements fast truncated SVD (Singular Value Decomposition). The SVD decomposition can be updated with new observations
at any time, for an online, incremental, memory-efficient training.

This module actually contains several algorithms for decomposition of large corpora, a
combination of which effectively and transparently allows building LSI models for:

* corpora much larger than RAM: only constant memory is needed, independent of
  the corpus size
* corpora that are streamed: documents are only accessed sequentially, no
  random access
* corpora that cannot be even temporarily stored: each document can only be
  seen once and must be processed immediately (one-pass algorithm)
* distributed computing for very large corpora, making use of a cluster of
  machines

Wall-clock `performance on the English Wikipedia <https://radimrehurek.com/gensim/wiki.html>`_
(2G corpus positions, 3.2M documents, 100K features, 0.5G non-zero entries in the final TF-IDF matrix),
requesting the top 400 LSI factors:

====================================================== ============ ==================
 algorithm                                             serial       distributed
====================================================== ============ ==================
 one-pass merge algorithm                              5h14m        1h41m
 multi-pass stochastic algo (with 2 power iterations)  5h39m        N/A [1]_
====================================================== ============ ==================


*serial* = Core 2 Duo MacBook Pro 2.53Ghz, 4GB RAM, libVec

*distributed* = cluster of four logical nodes on three physical machines, each
with dual core Xeon 2.0GHz, 4GB RAM, ATLAS


Examples
--------
.. sourcecode:: pycon

    >>> from gensim.test.utils import common_dictionary, common_corpus
    >>> from gensim.models import LsiModel
    >>>
    >>> model = LsiModel(common_corpus, id2word=common_dictionary)
    >>> vectorized_corpus = model[common_corpus]  # vectorize input copus in BoW format


.. [1] The stochastic algo could be distributed too, but most time is already spent
   reading/decompressing the input from disk in its 4 passes. The extra network
   traffic due to data distribution across cluster nodes would likely make it
   *slower*.

"""

import logging
import sys
import time

import numpy as np
import scipy.linalg
import scipy.sparse
from scipy.sparse import sparsetools

from gensim import interfaces, matutils, utils
from gensim.models import basemodel
from gensim.utils import is_empty

logger = logging.getLogger(__name__)

# accuracy defaults for the multi-pass stochastic algo
P2_EXTRA_DIMS = 100  # set to `None` for dynamic P2_EXTRA_DIMS=k
P2_EXTRA_ITERS = 2


def clip_spectrum(s, k, discard=0.001):
    """Find how many factors should be kept to avoid storing spurious (tiny, numerically unstable) values.

    Parameters
    ----------
    s : list of float
        Eigenvalues of the original matrix.
    k : int
        Maximum desired rank (number of factors)
    discard: float
        Percentage of the spectrum's energy to be discarded.

    Returns
    -------
    int
        Rank (number of factors) of the reduced matrix.


    """
    # compute relative contribution of eigenvalues towards the energy spectrum
    rel_spectrum = np.abs(1.0 - np.cumsum(s / np.sum(s)))
    # ignore the last `discard` mass (or 1/k, whichever is smaller) of the spectrum
    small = 1 + len(np.where(rel_spectrum > min(discard, 1.0 / k))[0])
    k = min(k, small)  # clip against k
    logger.info("keeping %i factors (discarding %.3f%% of energy spectrum)", k, 100 * rel_spectrum[k - 1])
    return k


def asfarray(a, name=''):
    """Get an array laid out in Fortran order in memory.

    Parameters
    ----------
    a : numpy.ndarray
        Input array.
    name : str, optional
        Array name, used only for logging purposes.

    Returns
    -------
    np.ndarray
        The input `a` in Fortran, or column-major order.

    """
    if not a.flags.f_contiguous:
        logger.debug("converting %s array %s to FORTRAN order", a.shape, name)
        a = np.asfortranarray(a)
    return a


def ascarray(a, name=''):
    """Return a contiguous array in memory (C order).

    Parameters
    ----------
    a : numpy.ndarray
        Input array.
    name : str, optional
        Array name, used for logging purposes.

    Returns
    -------
    np.ndarray
        Contiguous array (row-major order) of same shape and content as `a`.

    """
    if not a.flags.contiguous:
        logger.debug("converting %s array %s to C order", a.shape, name)
        a = np.ascontiguousarray(a)
    return a


class Projection(utils.SaveLoad):
    """Low dimensional projection of a term-document matrix.

    This is the class taking care of the 'core math': interfacing with corpora, splitting large corpora into chunks
    and merging them etc. This done through the higher-level :class:`~gensim.models.lsimodel.LsiModel` class.

    Notes
    -----
    The projection can be later updated by merging it with another :class:`~gensim.models.lsimodel.Projection`
    via  :meth:`~gensim.models.lsimodel.Projection.merge`. This is how incremental training actually happens.

    """

    def __init__(
            self, m, k, docs=None, use_svdlibc=False, power_iters=P2_EXTRA_ITERS,
            extra_dims=P2_EXTRA_DIMS, dtype=np.float64, random_seed=None,
    ):
        """Construct the (U, S) projection from a corpus.

        Parameters
        ----------
        m : int
            Number of features (terms) in the corpus.
        k : int
            Desired rank of the decomposed matrix.
        docs : {iterable of list of (int, float), scipy.sparse.csc}
            Corpus in BoW format or as sparse matrix.
        use_svdlibc : bool, optional
            If True - will use `sparsesvd library <https://pypi.org/project/sparsesvd/>`_,
            otherwise - our own version will be used.
        power_iters: int, optional
            Number of power iteration steps to be used. Tune to improve accuracy.
        extra_dims : int, optional
            Extra samples to be used besides the rank `k`. Tune to improve accuracy.
        dtype : numpy.dtype, optional
            Enforces a type for elements of the decomposed matrix.
        random_seed: {None, int}, optional
            Random seed used to initialize the pseudo-random number generator,
            a local instance of numpy.random.RandomState instance.

        """
        self.m, self.k = m, k
        self.power_iters = power_iters
        self.extra_dims = extra_dims
        self.random_seed = random_seed
        if docs is not None:
            # base case decomposition: given a job `docs`, compute its decomposition,
            # *in-core*.
            if not use_svdlibc:
                u, s = stochastic_svd(
                    docs, k, chunksize=sys.maxsize,
                    num_terms=m, power_iters=self.power_iters,
                    extra_dims=self.extra_dims, dtype=dtype, random_seed=self.random_seed)
            else:
                try:
                    import sparsesvd
                except ImportError:
                    raise ImportError("`sparsesvd` module requested but not found; run `easy_install sparsesvd`")
                logger.info("computing sparse SVD of %s matrix", str(docs.shape))
                if not scipy.sparse.issparse(docs):
                    docs = matutils.corpus2csc(docs)
                # ask for extra factors, because for some reason SVDLIBC sometimes returns fewer factors than requested
                ut, s, vt = sparsesvd.sparsesvd(docs, k + 30)
                u = ut.T
                del ut, vt
                k = clip_spectrum(s ** 2, self.k)
            self.u = u[:, :k].copy()
            self.s = s[:k].copy()
        else:
            self.u, self.s = None, None

    def empty_like(self):
        """Get an empty Projection with the same parameters as the current object.

        Returns
        -------
        :class:`~gensim.models.lsimodel.Projection`
            An empty copy (without corpus) of the current projection.

        """
        return Projection(
            self.m, self.k, power_iters=self.power_iters,
            extra_dims=self.extra_dims, random_seed=self.random_seed,
        )

    def merge(self, other, decay=1.0):
        """Merge current :class:`~gensim.models.lsimodel.Projection` instance with another.

        Warnings
        --------
        The content of `other` is destroyed in the process, so pass this function a copy of `other`
        if you need it further. The `other` :class:`~gensim.models.lsimodel.Projection` is expected to contain
        the same number of features.

        Parameters
        ----------
        other : :class:`~gensim.models.lsimodel.Projection`
            The Projection object to be merged into the current one. It will be destroyed after merging.
        decay : float, optional
            Weight of existing observations relatively to new ones.
            Setting `decay` < 1.0 causes re-orientation towards new data trends in the input document stream,
            by giving less emphasis to old observations. This allows LSA to gradually "forget" old observations
            (documents) and give more preference to new ones.

        """
        if other.u is None:
            # the other projection is empty => do nothing
            return
        if self.u is None:
            # we are empty => result of merge is the other projection, whatever it is
            self.u = other.u.copy()
            self.s = other.s.copy()
            return
        if self.m != other.m:
            raise ValueError(
                "vector space mismatch: update is using %s features, expected %s" % (other.m, self.m)
            )
        logger.info("merging projections: %s + %s", str(self.u.shape), str(other.u.shape))
        m, n1, n2 = self.u.shape[0], self.u.shape[1], other.u.shape[1]
        # TODO Maybe keep the bases as elementary reflectors, without
        # forming explicit matrices with ORGQR.
        # The only operation we ever need is basis^T*basis ond basis*component.
        # But how to do that in scipy? And is it fast(er)?

        # find component of u2 orthogonal to u1
        logger.debug("constructing orthogonal component")
        self.u = asfarray(self.u, 'self.u')
        c = np.dot(self.u.T, other.u)
        self.u = ascarray(self.u, 'self.u')
        other.u -= np.dot(self.u, c)

        other.u = [other.u]  # do some reference magic and call qr_destroy, to save RAM
        q, r = matutils.qr_destroy(other.u)  # q, r = QR(component)
        assert not other.u

        # find the rotation that diagonalizes r
        k = np.bmat([
            [np.diag(decay * self.s), np.multiply(c, other.s)],
            [matutils.pad(np.array([]).reshape(0, 0), min(m, n2), n1), np.multiply(r, other.s)]
        ])
        logger.debug("computing SVD of %s dense matrix", k.shape)
        try:
            # in np < 1.1.0, running SVD sometimes results in "LinAlgError: SVD did not converge'.
            # for these early versions of np, catch the error and try to compute
            # SVD again, but over k*k^T.
            # see http://www.mail-archive.com/np-discussion@scipy.org/msg07224.html and
            # bug ticket http://projects.scipy.org/np/ticket/706
            # sdoering: replaced np's linalg.svd with scipy's linalg.svd:

            # TODO *ugly overkill*!! only need first self.k SVD factors... but there is no LAPACK wrapper
            # for partial svd/eigendecomp in np :( //sdoering: maybe there is one in scipy?
            u_k, s_k, _ = scipy.linalg.svd(k, full_matrices=False)
        except scipy.linalg.LinAlgError:
            logger.error("SVD(A) failed; trying SVD(A * A^T)")
            # if this fails too, give up with an exception
            u_k, s_k, _ = scipy.linalg.svd(np.dot(k, k.T), full_matrices=False)
            s_k = np.sqrt(s_k)  # go back from eigen values to singular values

        k = clip_spectrum(s_k ** 2, self.k)
        u1_k, u2_k, s_k = np.array(u_k[:n1, :k]), np.array(u_k[n1:, :k]), s_k[:k]

        # update & rotate current basis U = [U, U']*[U1_k, U2_k]
        logger.debug("updating orthonormal basis U")
        self.s = s_k
        self.u = ascarray(self.u, 'self.u')
        self.u = np.dot(self.u, u1_k)

        q = ascarray(q, 'q')
        q = np.dot(q, u2_k)
        self.u += q

        # make each column of U start with a non-negative number (to force canonical decomposition)
        if self.u.shape[0] > 0:
            for i in range(self.u.shape[1]):
                if self.u[0, i] < 0.0:
                    self.u[:, i] *= -1.0


class LsiModel(interfaces.TransformationABC, basemodel.BaseTopicModel):
    """Model for `Latent Semantic Indexing
    <https://en.wikipedia.org/wiki/Latent_semantic_analysis#Latent_semantic_indexing>`_.

    The decomposition algorithm is described in `"Fast and Faster: A Comparison of Two Streamed
    Matrix Decomposition Algorithms" <https://arxiv.org/pdf/1102.5597.pdf>`_.

    Notes
    -----
    * :attr:`gensim.models.lsimodel.LsiModel.projection.u` - left singular vectors,
    * :attr:`gensim.models.lsimodel.LsiModel.projection.s` - singular values,
    * ``model[training_corpus]`` - right singular vectors (can be reconstructed if needed).

    See Also
    --------
    `FAQ about LSI matrices
    <https://github.com/RaRe-Technologies/gensim/wiki/Recipes-&-FAQ#q4-how-do-you-output-the-u-s-vt-matrices-of-lsi>`_.

    Examples
    --------
    .. sourcecode:: pycon

        >>> from gensim.test.utils import common_corpus, common_dictionary, get_tmpfile
        >>> from gensim.models import LsiModel
        >>>
        >>> model = LsiModel(common_corpus[:3], id2word=common_dictionary)  # train model
        >>> vector = model[common_corpus[4]]  # apply model to BoW document
        >>> model.add_documents(common_corpus[4:])  # update model with new documents
        >>> tmp_fname = get_tmpfile("lsi.model")
        >>> model.save(tmp_fname)  # save model
        >>> loaded_model = LsiModel.load(tmp_fname)  # load model

    """

    def __init__(
            self, corpus=None, num_topics=200, id2word=None, chunksize=20000,
            decay=1.0, distributed=False, onepass=True, power_iters=P2_EXTRA_ITERS,
            extra_samples=P2_EXTRA_DIMS, dtype=np.float64, random_seed=None,
    ):
        """Build an LSI model.

        Parameters
        ----------
        corpus : {iterable of list of (int, float), scipy.sparse.csc}, optional
            Stream of document vectors or a sparse matrix of shape (`num_documents`, `num_terms`).
        num_topics : int, optional
            Number of requested factors (latent dimensions)
        id2word : dict of {int: str}, optional
            ID to word mapping, optional.
        chunksize :  int, optional
            Number of documents to be used in each training chunk.
        decay : float, optional
            Weight of existing observations relatively to new ones.
        distributed : bool, optional
            If True - distributed mode (parallel execution on several machines) will be used.
        onepass : bool, optional
            Whether the one-pass algorithm should be used for training.
            Pass `False` to force a multi-pass stochastic algorithm.
        power_iters: int, optional
            Number of power iteration steps to be used.
            Increasing the number of power iterations improves accuracy, but lowers performance
        extra_samples : int, optional
            Extra samples to be used besides the rank `k`. Can improve accuracy.
        dtype : type, optional
            Enforces a type for elements of the decomposed matrix.
        random_seed: {None, int}, optional
            Random seed used to initialize the pseudo-random number generator,
            a local instance of numpy.random.RandomState instance.

        """
        self.id2word = id2word
        self.num_topics = int(num_topics)
        self.chunksize = int(chunksize)
        self.decay = float(decay)
        if distributed:
            if not onepass:
                logger.warning("forcing the one-pass algorithm for distributed LSA")
                onepass = True
        self.onepass = onepass
        self.extra_samples, self.power_iters = extra_samples, power_iters
        self.dtype = dtype
        self.random_seed = random_seed

        if corpus is None and self.id2word is None:
            raise ValueError(
                'at least one of corpus/id2word must be specified, to establish input space dimensionality'
            )

        if self.id2word is None:
            logger.warning("no word id mapping provided; initializing from corpus, assuming identity")
            self.id2word = utils.dict_from_corpus(corpus)
            self.num_terms = len(self.id2word)
        else:
            self.num_terms = 1 + (max(self.id2word.keys()) if self.id2word else -1)

        self.docs_processed = 0
        self.projection = Projection(
            self.num_terms, self.num_topics, power_iters=self.power_iters,
            extra_dims=self.extra_samples, dtype=dtype, random_seed=self.random_seed
        )

        self.numworkers = 1
        if not distributed:
            logger.info("using serial LSI version on this node")
            self.dispatcher = None
        else:
            if not onepass:
                raise NotImplementedError(
                    "distributed stochastic LSA not implemented yet; "
                    "run either distributed one-pass, or serial randomized."
                )
            try:
                import Pyro4
                dispatcher = Pyro4.Proxy('PYRONAME:gensim.lsi_dispatcher')
                logger.debug("looking for dispatcher at %s", str(dispatcher._pyroUri))
                dispatcher.initialize(
                    id2word=self.id2word, num_topics=num_topics, chunksize=chunksize, decay=decay,
                    power_iters=self.power_iters, extra_samples=self.extra_samples, distributed=False, onepass=onepass
                )
                self.dispatcher = dispatcher
                self.numworkers = len(dispatcher.getworkers())
                logger.info("using distributed version with %i workers", self.numworkers)
            except Exception as err:
                # distributed version was specifically requested, so this is an error state
                logger.error("failed to initialize distributed LSI (%s)", err)
                raise RuntimeError("failed to initialize distributed LSI (%s)" % err)

        if corpus is not None:
            start = time.time()
            self.add_documents(corpus)
            self.add_lifecycle_event(
                "created",
                msg=f"trained {self} in {time.time() - start:.2f}s",
            )

    def add_documents(self, corpus, chunksize=None, decay=None):
        """Update model with new `corpus`.

        Parameters
        ----------
        corpus : {iterable of list of (int, float), scipy.sparse.csc}
            Stream of document vectors or sparse matrix of shape (`num_terms`, num_documents).
        chunksize : int, optional
            Number of documents to be used in each training chunk, will use `self.chunksize` if not specified.
        decay : float, optional
            Weight of existing observations relatively to new ones,  will use `self.decay` if not specified.

        Notes
        -----
        Training proceeds in chunks of `chunksize` documents at a time. The size of `chunksize` is a tradeoff
        between increased speed (bigger `chunksize`) vs. lower memory footprint (smaller `chunksize`).
        If the distributed mode is on, each chunk is sent to a different worker/computer.

        """
        logger.info("updating model with new documents")

        # get computation parameters; if not specified, use the ones from constructor
        if chunksize is None:
            chunksize = self.chunksize
        if decay is None:
            decay = self.decay
        if is_empty(corpus):
            logger.warning('LsiModel.add_documents() called but no documents provided, is this intended?')
        if not scipy.sparse.issparse(corpus):
            if not self.onepass:
                # we are allowed multiple passes over the input => use a faster, randomized two-pass algo
                update = Projection(
                    self.num_terms, self.num_topics, None,
                    dtype=self.dtype, random_seed=self.random_seed,
                )
                update.u, update.s = stochastic_svd(
                    corpus, self.num_topics,
                    num_terms=self.num_terms, chunksize=chunksize,
                    extra_dims=self.extra_samples, power_iters=self.power_iters, dtype=self.dtype,
                    random_seed=self.random_seed,
                )
                self.projection.merge(update, decay=decay)
                self.docs_processed += len(corpus) if hasattr(corpus, '__len__') else 0
            else:
                # the one-pass algo
                doc_no = 0
                if self.dispatcher:
                    logger.info('initializing %s workers', self.numworkers)
                    self.dispatcher.reset()
                for chunk_no, chunk in enumerate(utils.grouper(corpus, chunksize)):
                    logger.info("preparing a new chunk of documents")
                    nnz = sum(len(doc) for doc in chunk)
                    # construct the job as a sparse matrix, to minimize memory overhead
                    # definitely avoid materializing it as a dense matrix!
                    logger.debug("converting corpus to csc format")
                    job = matutils.corpus2csc(
                        chunk, num_docs=len(chunk), num_terms=self.num_terms,
                        num_nnz=nnz, dtype=self.dtype,
                    )
                    del chunk
                    doc_no += job.shape[1]
                    if self.dispatcher:
                        # distributed version: add this job to the job queue, so workers can work on it
                        logger.debug("creating job #%i", chunk_no)
                        # put job into queue; this will eventually block, because the queue has a small finite size
                        self.dispatcher.putjob(job)
                        del job
                        logger.info("dispatched documents up to #%s", doc_no)
                    else:
                        # serial version, there is only one "worker" (myself) => process the job directly
                        update = Projection(
                            self.num_terms, self.num_topics, job, extra_dims=self.extra_samples,
                            power_iters=self.power_iters, dtype=self.dtype, random_seed=self.random_seed,
                        )
                        del job
                        self.projection.merge(update, decay=decay)
                        del update
                        logger.info("processed documents up to #%s", doc_no)
                        self.print_topics(5)

                # wait for all workers to finish (distributed version only)
                if self.dispatcher:
                    logger.info("reached the end of input; now waiting for all remaining jobs to finish")
                    self.projection = self.dispatcher.getstate()
                self.docs_processed += doc_no
        else:
            assert not self.dispatcher, "must be in serial mode to receive jobs"
            update = Projection(
                self.num_terms, self.num_topics, corpus.tocsc(), extra_dims=self.extra_samples,
                power_iters=self.power_iters, dtype=self.dtype,
            )
            self.projection.merge(update, decay=decay)
            logger.info("processed sparse job of %i documents", corpus.shape[1])
            self.docs_processed += corpus.shape[1]

    def __str__(self):
        """Get a human readable representation of model.

        Returns
        -------
        str
            A human readable string of the current objects parameters.

        """
        return "%s<num_terms=%s, num_topics=%s, decay=%s, chunksize=%s>" % (
            self.__class__.__name__, self.num_terms, self.num_topics, self.decay, self.chunksize
        )

    def __getitem__(self, bow, scaled=False, chunksize=512):
        """Get the latent representation for `bow`.

        Parameters
        ----------
        bow : {list of (int, int), iterable of list of (int, int)}
            Document or corpus in BoW representation.
        scaled : bool, optional
            If True - topics will be scaled by the inverse of singular values.
        chunksize :  int, optional
            Number of documents to be used in each applying chunk.

        Returns
        -------
        list of (int, float)
            Latent representation of topics in BoW format for document **OR**
        :class:`gensim.matutils.Dense2Corpus`
            Latent representation of corpus in BoW format if `bow` is corpus.

        """
        if self.projection.u is None:
            raise ValueError('No training data provided - LSI model not initialized yet')

        # if the input vector is in fact a corpus, return a transformed corpus as a result
        is_corpus, bow = utils.is_corpus(bow)
        if is_corpus and chunksize:
            # by default, transform `chunksize` documents at once, when called as `lsi[corpus]`.
            # this chunking is completely transparent to the user, but it speeds
            # up internal computations (one mat * mat multiplication, instead of
            # `chunksize` smaller mat * vec multiplications).
            return self._apply(bow, chunksize=chunksize)

        if not is_corpus:
            bow = [bow]

        # convert input to scipy.sparse CSC, then do "sparse * dense = dense" multiplication
        vec = matutils.corpus2csc(bow, num_terms=self.num_terms, dtype=self.projection.u.dtype)
        topic_dist = (vec.T * self.projection.u[:, :self.num_topics]).T  # (x^T * u).T = u^-1 * x

        # # convert input to dense, then do dense * dense multiplication
        # # ± same performance as above (BLAS dense * dense is better optimized than scipy.sparse),
        # but consumes more memory
        # vec = matutils.corpus2dense(bow, num_terms=self.num_terms, num_docs=len(bow))
        # topic_dist = np.dot(self.projection.u[:, :self.num_topics].T, vec)

        # # use np's advanced indexing to simulate sparse * dense
        # # ± same speed again
        # u = self.projection.u[:, :self.num_topics]
        # topic_dist = np.empty((u.shape[1], len(bow)), dtype=u.dtype)
        # for vecno, vec in enumerate(bow):
        #     indices, data = zip(*vec) if vec else ([], [])
        #     topic_dist[:, vecno] = np.dot(u.take(indices, axis=0).T, np.array(data, dtype=u.dtype))

        if not is_corpus:
            # convert back from matrix into a 1d vec
            topic_dist = topic_dist.reshape(-1)

        if scaled:
            topic_dist = (1.0 / self.projection.s[:self.num_topics]) * topic_dist  # s^-1 * u^-1 * x

        # convert a np array to gensim sparse vector = tuples of (feature_id, feature_weight),
        # with no zero weights.
        if not is_corpus:
            # lsi[single_document]
            result = matutils.full2sparse(topic_dist)
        else:
            # lsi[chunk of documents]
            result = matutils.Dense2Corpus(topic_dist)
        return result

    def get_topics(self):
        """Get the topic vectors.

        Notes
        -----
        The number of topics can actually be smaller than `self.num_topics`, if there were not enough factors
        in the matrix (real rank of input matrix smaller than `self.num_topics`).

        Returns
        -------
        np.ndarray
            The term topic matrix with shape (`num_topics`, `vocabulary_size`)

        """
        projections = self.projection.u.T
        num_topics = len(projections)
        topics = []
        for i in range(num_topics):
            c = np.asarray(projections[i, :]).flatten()
            norm = np.sqrt(np.sum(np.dot(c, c)))
            topics.append(1.0 * c / norm)
        return np.array(topics)

    def show_topic(self, topicno, topn=10):
        """Get the words that define a topic along with their contribution.

        This is actually the left singular vector of the specified topic.

        The most important words in defining the topic (greatest absolute value) are included
        in the output, along with their contribution to the topic.

        Parameters
        ----------
        topicno : int
            The topics id number.
        topn : int
            Number of words to be included to the result.

        Returns
        -------
        list of (str, float)
            Topic representation in BoW format.

        """
        # size of the projection matrix can actually be smaller than `self.num_topics`,
        # if there were not enough factors (real rank of input matrix smaller than
        # `self.num_topics`). in that case, return an empty string
        if topicno >= len(self.projection.u.T):
            return ''
        c = np.asarray(self.projection.u.T[topicno, :]).flatten()
        norm = np.sqrt(np.sum(np.dot(c, c)))
        most = matutils.argsort(np.abs(c), topn, reverse=True)

        # Output only (word, score) pairs for `val`s that are within `self.id2word`.  See #3090 for details.
        return [(self.id2word[val], 1.0 * c[val] / norm) for val in most if val in self.id2word]

    def show_topics(self, num_topics=-1, num_words=10, log=False, formatted=True):
        """Get the most significant topics.

        Parameters
        ----------
        num_topics : int, optional
            The number of topics to be selected, if -1 - all topics will be in result (ordered by significance).
        num_words : int, optional
            The number of words to be included per topics (ordered by significance).
        log : bool, optional
            If True - log topics with logger.
        formatted : bool, optional
            If True - each topic represented as string, otherwise - in BoW format.

        Returns
        -------
        list of (int, str)
            If `formatted=True`, return sequence with (topic_id, string representation of topics) **OR**
        list of (int, list of (str, float))
            Otherwise, return sequence with (topic_id, [(word, value), ... ]).

        """
        shown = []
        if num_topics < 0:
            num_topics = self.num_topics
        for i in range(min(num_topics, self.num_topics)):
            if i < len(self.projection.s):
                if formatted:
                    topic = self.print_topic(i, topn=num_words)
                else:
                    topic = self.show_topic(i, topn=num_words)
                shown.append((i, topic))
                if log:
                    logger.info("topic #%i(%.3f): %s", i, self.projection.s[i], topic)
        return shown

    def print_debug(self, num_topics=5, num_words=10):
        """Print (to log) the most salient words of the first `num_topics` topics.

        Unlike :meth:`~gensim.models.lsimodel.LsiModel.print_topics`, this looks for words that are significant for
        a particular topic *and* not for others. This *should* result in a
        more human-interpretable description of topics.

        Alias for :func:`~gensim.models.lsimodel.print_debug`.

        Parameters
        ----------
        num_topics : int, optional
            The number of topics to be selected (ordered by significance).
        num_words : int, optional
            The number of words to be included per topics (ordered by significance).

        """
        # only wrap the module-level fnc
        print_debug(
            self.id2word, self.projection.u, self.projection.s,
            range(min(num_topics, len(self.projection.u.T))),
            num_words=num_words,
        )

    def save(self, fname, *args, **kwargs):
        """Save the model to a file.

        Notes
        -----
        Large internal arrays may be stored into separate files, with `fname` as prefix.

        Warnings
        --------
        Do not save as a compressed file if you intend to load the file back with `mmap`.

        Parameters
        ----------
        fname : str
            Path to output file.
        *args
            Variable length argument list, see :meth:`gensim.utils.SaveLoad.save`.
        **kwargs
            Arbitrary keyword arguments, see :meth:`gensim.utils.SaveLoad.save`.

        See Also
        --------
        :meth:`~gensim.models.lsimodel.LsiModel.load`

        """

        if self.projection is not None:
            self.projection.save(utils.smart_extension(fname, '.projection'), *args, **kwargs)
        super(LsiModel, self).save(fname, *args, ignore=['projection', 'dispatcher'], **kwargs)

    @classmethod
    def load(cls, fname, *args, **kwargs):
        """Load a previously saved object using :meth:`~gensim.models.lsimodel.LsiModel.save` from file.

        Notes
        -----
        Large arrays can be memmap'ed back as read-only (shared memory) by setting the `mmap='r'` parameter.

        Parameters
        ----------
        fname : str
            Path to file that contains LsiModel.
        *args
            Variable length argument list, see :meth:`gensim.utils.SaveLoad.load`.
        **kwargs
            Arbitrary keyword arguments, see :meth:`gensim.utils.SaveLoad.load`.

        See Also
        --------
        :meth:`~gensim.models.lsimodel.LsiModel.save`

        Returns
        -------
        :class:`~gensim.models.lsimodel.LsiModel`
            Loaded instance.

        Raises
        ------
        IOError
            When methods are called on instance (should be called from class).

        """
        kwargs['mmap'] = kwargs.get('mmap', None)
        result = super(LsiModel, cls).load(fname, *args, **kwargs)
        projection_fname = utils.smart_extension(fname, '.projection')
        try:
            result.projection = super(LsiModel, cls).load(projection_fname, *args, **kwargs)
        except Exception as e:
            logging.warning("failed to load projection from %s: %s", projection_fname, e)
        return result


def print_debug(id2token, u, s, topics, num_words=10, num_neg=None):
    """Log the most salient words per topic.

    Parameters
    ----------
    id2token : :class:`~gensim.corpora.dictionary.Dictionary`
        Mapping from ID to word in the Dictionary.
    u : np.ndarray
        The 2D U decomposition matrix.
    s : np.ndarray
        The 1D reduced array of eigenvalues used for decomposition.
    topics : list of int
        Sequence of topic IDs to be printed
    num_words : int, optional
        Number of words to be included for each topic.
    num_neg : int, optional
        Number of words with a negative contribution to a topic that should be included.

    """
    if num_neg is None:
        # by default, print half as many salient negative words as positive
        num_neg = num_words / 2

    logger.info('computing word-topic salience for %i topics', len(topics))
    topics, result = set(topics), {}
    # TODO speed up by block computation
    for uvecno, uvec in enumerate(u):
        uvec = np.abs(np.asarray(uvec).flatten())
        udiff = uvec / np.sqrt(np.sum(np.dot(uvec, uvec)))
        for topic in topics:
            result.setdefault(topic, []).append((udiff[topic], uvecno))

    logger.debug("printing %i+%i salient words", num_words, num_neg)
    for topic in sorted(result.keys()):
        weights = sorted(result[topic], key=lambda x: -abs(x[0]))
        _, most = weights[0]
        if u[most, topic] < 0.0:  # the most significant word has a negative sign => flip sign of u[most]
            normalize = -1.0
        else:
            normalize = 1.0

        # order features according to salience; ignore near-zero entries in u
        pos, neg = [], []
        for weight, uvecno in weights:
            if normalize * u[uvecno, topic] > 0.0001:
                pos.append('%s(%.3f)' % (id2token[uvecno], u[uvecno, topic]))
                if len(pos) >= num_words:
                    break

        for weight, uvecno in weights:
            if normalize * u[uvecno, topic] < -0.0001:
                neg.append('%s(%.3f)' % (id2token[uvecno], u[uvecno, topic]))
                if len(neg) >= num_neg:
                    break

        logger.info('topic #%s(%.3f): %s, ..., %s', topic, s[topic], ', '.join(pos), ', '.join(neg))


def stochastic_svd(
        corpus, rank, num_terms, chunksize=20000, extra_dims=None,
        power_iters=0, dtype=np.float64, eps=1e-6, random_seed=None,
    ):
    """Run truncated Singular Value Decomposition (SVD) on a sparse input.

    Parameters
    ----------
    corpus : {iterable of list of (int, float), scipy.sparse}
        Input corpus as a stream (does not have to fit in RAM)
        or a sparse matrix of shape (`num_terms`, num_documents).
    rank : int
        Desired number of factors to be retained after decomposition.
    num_terms : int
        The number of features (terms) in `corpus`.
    chunksize :  int, optional
        Number of documents to be used in each training chunk.
    extra_dims : int, optional
        Extra samples to be used besides the rank `k`. Can improve accuracy.
    power_iters: int, optional
        Number of power iteration steps to be used. Increasing the number of power iterations improves accuracy,
        but lowers performance.
    dtype : numpy.dtype, optional
        Enforces a type for elements of the decomposed matrix.
    eps: float, optional
        Percentage of the spectrum's energy to be discarded.
    random_seed: {None, int}, optional
        Random seed used to initialize the pseudo-random number generator,
         a local instance of numpy.random.RandomState instance.


    Notes
    -----
    The corpus may be larger than RAM (iterator of vectors), if `corpus` is a `scipy.sparse.csc` instead,
    it is assumed the whole corpus fits into core memory and a different (more efficient) code path is chosen.
    This may return less than the requested number of top `rank` factors, in case the input itself is of lower rank.
    The `extra_dims` (oversampling) and especially `power_iters` (power iterations) parameters affect accuracy of the
    decomposition.

    This algorithm uses `2 + power_iters` passes over the input data. In case you can only afford a single pass,
    set `onepass=True` in :class:`~gensim.models.lsimodel.LsiModel` and avoid using this function directly.

    The decomposition algorithm is based on `"Finding structure with randomness:
    Probabilistic algorithms for constructing approximate matrix decompositions" <https://arxiv.org/abs/0909.4061>`_.


    Returns
    -------
    (np.ndarray 2D, np.ndarray 1D)
        The left singular vectors and the singular values of the `corpus`.

    """
    rank = int(rank)
    if extra_dims is None:
        samples = max(10, 2 * rank)  # use more samples than requested factors, to improve accuracy
    else:
        samples = rank + int(extra_dims)
    logger.info("using %i extra samples and %i power iterations", samples - rank, power_iters)

    num_terms = int(num_terms)

    # first phase: construct the orthonormal action matrix Q = orth(Y) = orth((A * A.T)^q * A * O)
    # build Y in blocks of `chunksize` documents (much faster than going one-by-one
    # and more memory friendly than processing all documents at once)
    y = np.zeros(dtype=dtype, shape=(num_terms, samples))
    logger.info("1st phase: constructing %s action matrix", str(y.shape))
    random_state = np.random.RandomState(random_seed)

    if scipy.sparse.issparse(corpus):
        m, n = corpus.shape
        assert num_terms == m, f"mismatch in number of features: {m} in sparse matrix vs. {num_terms} parameter"
        o = random_state.normal(0.0, 1.0, (n, samples)).astype(y.dtype)  # draw a random gaussian matrix
        sparsetools.csc_matvecs(
            m, n, samples, corpus.indptr, corpus.indices,
            corpus.data, o.ravel(), y.ravel(),
        )  # y = corpus * o
        del o

        # unlike np, scipy.sparse `astype()` copies everything, even if there is no change to dtype!
        # so check for equal dtype explicitly, to avoid the extra memory footprint if possible
        if y.dtype != dtype:
            y = y.astype(dtype)

        logger.info("orthonormalizing %s action matrix", str(y.shape))
        y = [y]
        q, _ = matutils.qr_destroy(y)  # orthonormalize the range

        logger.debug("running %i power iterations", power_iters)
        for _ in range(power_iters):
            q = corpus.T * q
            q = [corpus * q]
            q, _ = matutils.qr_destroy(q)  # orthonormalize the range after each power iteration step
    else:
        num_docs = 0
        for chunk_no, chunk in enumerate(utils.grouper(corpus, chunksize)):
            logger.info('PROGRESS: at document #%i', (chunk_no * chunksize))
            # construct the chunk as a sparse matrix, to minimize memory overhead
            # definitely avoid materializing it as a dense (num_terms x chunksize) matrix!
            s = sum(len(doc) for doc in chunk)
            chunk = matutils.corpus2csc(chunk, num_terms=num_terms, dtype=dtype)  # documents = columns of sparse CSC
            m, n = chunk.shape
            assert m == num_terms
            assert n <= chunksize  # the very last chunk of A is allowed to be smaller in size
            num_docs += n
            logger.debug("multiplying chunk * gauss")
            o = random_state.normal(0.0, 1.0, (n, samples), ).astype(dtype)  # draw a random gaussian matrix
            sparsetools.csc_matvecs(
                m, n, samples, chunk.indptr, chunk.indices,  # y = y + chunk * o
                chunk.data, o.ravel(), y.ravel(),
            )
            del chunk, o
        y = [y]
        q, _ = matutils.qr_destroy(y)  # orthonormalize the range

        for power_iter in range(power_iters):
            logger.info("running power iteration #%i", power_iter + 1)
            yold = q.copy()
            q[:] = 0.0
            for chunk_no, chunk in enumerate(utils.grouper(corpus, chunksize)):
                logger.info('PROGRESS: at document #%i/%i', chunk_no * chunksize, num_docs)
                # documents = columns of sparse CSC
                chunk = matutils.corpus2csc(chunk, num_terms=num_terms, dtype=dtype)
                tmp = chunk.T * yold
                tmp = chunk * tmp
                del chunk
                q += tmp
            del yold
            q = [q]
            q, _ = matutils.qr_destroy(q)  # orthonormalize the range

    qt = q[:, :samples].T.copy()
    del q

    if scipy.sparse.issparse(corpus):
        b = qt * corpus
        logger.info("2nd phase: running dense svd on %s matrix", str(b.shape))
        u, s, vt = scipy.linalg.svd(b, full_matrices=False)
        del b, vt
    else:
        # second phase: construct the covariance matrix X = B * B.T, where B = Q.T * A
        # again, construct X incrementally, in chunks of `chunksize` documents from the streaming
        # input corpus A, to avoid using O(number of documents) memory
        x = np.zeros(shape=(qt.shape[0], qt.shape[0]), dtype=dtype)
        logger.info("2nd phase: constructing %s covariance matrix", str(x.shape))
        for chunk_no, chunk in enumerate(utils.grouper(corpus, chunksize)):
            logger.info('PROGRESS: at document #%i/%i', chunk_no * chunksize, num_docs)
            chunk = matutils.corpus2csc(chunk, num_terms=num_terms, dtype=qt.dtype)
            b = qt * chunk  # dense * sparse matrix multiply
            del chunk
            x += np.dot(b, b.T)  # TODO should call the BLAS routine SYRK, but there is no SYRK wrapper in scipy :(
            del b

        # now we're ready to compute decomposition of the small matrix X
        logger.info("running dense decomposition on %s covariance matrix", str(x.shape))
        # could use linalg.eigh, but who cares... and svd returns the factors already sorted :)
        u, s, vt = scipy.linalg.svd(x)
        # sqrt to go back from singular values of X to singular values of B = singular values of the corpus
        s = np.sqrt(s)
    q = qt.T.copy()
    del qt

    logger.info("computing the final decomposition")
    keep = clip_spectrum(s ** 2, rank, discard=eps)
    u = u[:, :keep].copy()
    s = s[:keep]
    u = np.dot(q, u)
    return u.astype(dtype), s.astype(dtype)