tl; dr: A simplified analysis of the article, in which the author offers two interesting theorems, on the basis of which he found a way to extract hidden vectors of meanings from the embedding matrix. A guide is given on how to reproduce the results. The laptop is available on github .

Introduction

In this article, I want to share one awesome thing that researcher Sanjev Arora found in his article Linear Algebraic Structure of Word Senses, with Applications to Polysemy . It is one of a series of articles in which he tries to provide a theoretical basis for the properties of word embeddings. In the same work, Arora makes the assumption that simple embeddings, such as word2vec or Glove, actually include several meanings for one word and offer a way to restore them. In the course of the article I will try to stick to the original examples.

More formally, for $$$\upsilon_{tie}$we denote a certain embedding vector of the word tie , which may have the meaning of a knot or tie, or it may be the verb “tie up”. Arora suggests that this vector can be written as the following linear combination

$$

$$\upsilon_{tie} \approx \alpha_1 \upsilon_{tie1} + \alpha_2 \upsilon_{tie2} + \alpha_3 \upsilon_{tie3}+...$$

Where $$$\upsilon_{tien}$this is one of the possible meanings of the word tie , and$$$\alpha$- coefficient. Let's try to figure out how this happens.

Theory

A short note on the theory of Arara

Since the starting work of Arara is much more complicated than this, I have not yet fully prepared the review. However, we will briefly see what it is.

So, Arora proposes the idea that any text is generated by a generative model. In the process of her work at every time step$$$t$ a word is generated $$$w$... The model consists of a context vector$$ and vectors of embeddings $$$u_w$. (dimensions), , . , , - (, ), — (, ), , , — .

, .. - , . . , . : " " , " ". , "": , .

, . , , , .

: , . , $$$t$ $$$w$

$$

$$P(w|c_t) = \frac{1}{Z_c}\exp{<c_t, \upsilon_w>}$$

$$$c_t$ — $$$t$, $$$\upsilon_w$ — $$$w$, $$$Z_c = \sum_w\exp<c,\upsilon_w>$ — partition function. , , .

. , , : , , , . Y, X .



. - , - .



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, ", , , ". , , , ", , , " , " " .

, . , , . , ( , ). , , . .

1

, $$$s$ $$$n$ . $$$A$ ,

$$

$$\upsilon_w \approx A \mathbb{E}[\frac{1}{n}\sum_{w_i\in s}\upsilon_{w_i}|w \in s]$$

, . . $$$w$ . $$$S$. , $$$\upsilon_s$ $$$s \in S$, $$$u$. , , $$$u$ $$$\upsilon_w$ $$$A$ ( ). , , out-of-vocabulary , , .

, . , SIF . , , , . , SIF $$$\upsilon_{SIF}$ k, , $$$w$, TF-IDF.

$$

$$\upsilon_{SIF}=\frac{1}{k}\sum^k_{n=1}\upsilon_n*tf\_idf(w_n)$$

, , 1, $$$c$. , - , , .

. , - $$$w$, $$$\upsilon_w$ , . :

1. . $$$V$.
2. $$$w' \in V$, , SIF 20 $$$w'$, . $$$w' \in V$ $$$(\nu_{w'}^1, \nu_{w'}^2,, ... \nu_{w'}^n,)$, n — $$$w'$ .
3. $$$u_{w'}$ SIF $$$w' \in V$ $$$u_{w'}=\frac{1}{n}\sum_{t=1}^n\nu_{w'}^t$.
4. $$$argmin_A\sum_A||Au_{w'}-\upsilon_{w'}||^2_2$
5. SIF $$$\upsilon_w=Au_w$

, .. . 1/3 , A 2\3 . . .

2

, $$$w$ $$$s_1$ $$$s_2$. $$< / m a t h - - > $ and $ < ! - - m a t h > \ upsilon$\upsilon_w$ - , . , , .. , , tie_1 tie_2, tie_1 — , tie2 — .

, , $<!-- math>$inline$ \upsilon{w{s1} } $</math -->$ $<!-- math>${w_{s2} } $inline$</math -->$. , , , $$$\lVert\upsilon_w - \upsilon^- \rVert \rightarrow 0$, $$$\upsilon^-$

$$

$$\upsilon_w^- = \frac{f_1}{f_1+f_2} * \upsilon_{s1} + \frac{f_2}{f_1+f_2} * \upsilon_{s2} = \alpha\upsilon_{s1} + \beta\upsilon_{s2}$$

$$$f_1$ $$$f_2$ $$$s_1$ and $$$s_2$ . , , .

, , , , ? , $$$alpha$. . , $$$c$ . , , . , , , , , . , , , (inner product) . , , - (, , , ), $$$\upsilon_{tie1}$ , ! .

. ? $$$\Re^d$ $$$k, n$. $$$k < n$, $$$A_1, A_2, ..., A_m$, ,

$$

$$\upsilon_w = \sum_{j=1}^m\alpha_{w,j}A_j + \mu_w$$

$$$k$ $$$\alpha$ $$$\mu_w$ — .

$$

$$\sum_w \lVert\upsilon_w - \sum_{j=1}^m\alpha_{w,j}A_j \rVert^2_2$$

, k (sparsity parameter), m — .. , . k-SVD. , . , $$$A$ , ( , $$$A$ ). , , - $$$A_i$ , , , $$$m$ . .

, , .

import numpy as np

from gensim.test.utils import datapath, get_tmpfile
from gensim.models import KeyedVectors
from gensim.scripts.glove2word2vec import glove2word2vec
from scipy.spatial.distance import cosine
import warnings
warnings.filterwarnings('ignore')

1. Gensim

GloVe.

, 300- .

tmp_file = get_tmpfile("test_word2vec.txt")
_ = glove2word2vec("/home/astromis/Embeddings/glove.6B.300d.txt", tmp_file)
model = KeyedVectors.load_word2vec_format(tmp_file)

embeddings = model.wv

index2word = embeddings.index2word
embedds = embeddings.vectors

print(embedds.shape)

(400000, 300)

400000 .

2. k-svd

. ksvd.

!pip install ksvd
from ksvd import ApproximateKSVD

Requirement already satisfied: ksvd in /home/astromis/anaconda3/lib/python3.6/site-packages (0.0.3)
Requirement already satisfied: numpy in /home/astromis/anaconda3/lib/python3.6/site-packages (from ksvd) (1.14.5)
Requirement already satisfied: scikit-learn in /home/astromis/anaconda3/lib/python3.6/site-packages (from ksvd) (0.19.1)

, 2000 5.

: 10000 . , , , , .

%time
aksvd = ApproximateKSVD(n_components=2000,transform_n_nonzero_coefs=5, )
embedding_trans = embeddings.vectors
dictionary = aksvd.fit(embedding_trans).components_
gamma = aksvd.transform(embedding_trans)

CPU times: user 4 µs, sys: 0 ns, total: 4 µs
Wall time: 9.54 µs

#gamma = np.load('./data/mats/.npz')
# dictionary_glove6b_300d.np.npz - whole matrix file
dictionary = np.load('./data/mats/dictionary_glove6b_300d_10000.np.npz')
dictionary = dictionary[dictionary.keys()[0]]

#print(gamma.shape)
print(dictionary.shape)

(2000, 300)

#np.savez_compressed('gamma_glove6b_300d.npz', gamma)
#np.savez_compressed('dictionary_glove6b_300d.npz', dictionary)

3.

, . .

embeddings.similar_by_vector(dictionary[1354,:])

[('slave', 0.8417330980300903),
 ('slaves', 0.7482961416244507),
 ('plantation', 0.6208109259605408),
 ('slavery', 0.5356900095939636),
 ('enslaved', 0.4814416170120239),
 ('indentured', 0.46423888206481934),
 ('fugitive', 0.4226764440536499),
 ('laborers', 0.41914862394332886),
 ('servitude', 0.41276970505714417),
 ('plantations', 0.4113745093345642)]

embeddings.similar_by_vector(dictionary[1350,:])

[('transplant', 0.7767853736877441),
 ('marrow', 0.699995219707489),
 ('transplants', 0.6998592615127563),
 ('kidney', 0.6526087522506714),
 ('transplantation', 0.6381147503852844),
 ('tissue', 0.6344675421714783),
 ('liver', 0.6085026860237122),
 ('blood', 0.5676015615463257),
 ('heart', 0.5653558969497681),
 ('cells', 0.5476219058036804)]

embeddings.similar_by_vector(dictionary[1546,:])

[('commons', 0.7160810828208923),
 ('house', 0.6588335037231445),
 ('parliament', 0.5054076910018921),
 ('capitol', 0.5014163851737976),
 ('senate', 0.4895153343677521),
 ('hill', 0.48859673738479614),
 ('inn', 0.4566132128238678),
 ('congressional', 0.4341348707675934),
 ('congress', 0.42997264862060547),
 ('parliamentary', 0.4264637529850006)]

embeddings.similar_by_vector(dictionary[1850,:])

[('okano', 0.2669774889945984),
 ('erythrocytes', 0.25755012035369873),
 ('windir', 0.25621023774147034),
 ('reapportionment', 0.2507009208202362),
 ('qurayza', 0.2459488958120346),
 ('taschen', 0.24417680501937866),
 ('pfaffenbach', 0.2437630295753479),
 ('boldt', 0.2394050508737564),
 ('frucht', 0.23922981321811676),
 ('rulebook', 0.23821482062339783)]

! , . . , , . "tie" "spring" .

itie = index2word.index('tie')
ispring = index2word.index('spring')

tie_emb = embedds[itie]
string_emb = embedds[ispring]

simlist = []

for i, vector in enumerate(dictionary):
    simlist.append( (cosine(vector, tie_emb), i) )

simlist = sorted(simlist, key=lambda x: x[0])
six_atoms_ind = [ins[1] for ins in simlist[:15]]

for atoms_idx in six_atoms_ind:
    nearest_words = embeddings.similar_by_vector(dictionary[atoms_idx,:])
    nearest_words = [word[0] for word in nearest_words]
    print("Atom #{}: {}".format(atoms_idx, ' '.join(nearest_words)))

Atom #162: win victory winning victories wins won 2-1 scored 3-1 scoring
Atom #58: game play match matches games played playing tournament players stadium
Atom #237: 0-0 1-1 2-2 3-3 draw 0-1 4-4 goalless 1-0 1-2
Atom #622: wrapped wrap wrapping holding placed attached tied hold plastic held
Atom #1899: struggles tying tied inextricably fortunes struggling tie intertwined redefine define
Atom #1941: semifinals quarterfinals semifinal quarterfinal finals semis semi-finals berth champions quarter-finals
Atom #1074: qualifier quarterfinals semifinal semifinals semi finals quarterfinal champion semis champions
Atom #1914: wearing wore jacket pants dress wear worn trousers shirt jeans
Atom #281: black wearing man pair white who girl young woman big
Atom #1683: overtime extra seconds ot apiece 20-17 turnovers 3-2 halftime overtimes
Atom #369: snap picked snapped pick grabbed picks knocked picking bounced pulled
Atom #98: first team start final second next time before test after
Atom #1455: after later before when then came last took again but
Atom #1203: competitions qualifying tournaments finals qualification matches qualifiers champions competition competed
Atom #1602: hat hats mask trick wearing wears sunglasses trademark wig wore

simlist = []

for i, vector in enumerate(dictionary):
    simlist.append( (cosine(vector, string_emb), i) )

simlist = sorted(simlist, key=lambda x: x[0])
six_atoms_ind = [ins[1] for ins in simlist[:15]]

for atoms_idx in six_atoms_ind:
    nearest_words = embeddings.similar_by_vector(dictionary[atoms_idx,:])
    nearest_words = [word[0] for word in nearest_words]
    print("Atom #{}: {}".format(atoms_idx, ' '.join(nearest_words)))

Atom #528: autumn spring summer winter season rainy seasons fall seasonal during
Atom #1070: start begin beginning starting starts begins next coming day started
Atom #931: holiday christmas holidays easter thanksgiving eve celebrate celebrations weekend festivities
Atom #1455: after later before when then came last took again but
Atom #754: but so not because even only that it this they
Atom #688: yankees yankee mets sox baseball braves steinbrenner dodgers orioles torre
Atom #1335: last ago year months years since month weeks week has
Atom #252: upcoming scheduled preparations postponed slated forthcoming planned delayed preparation preparing
Atom #619: cold cool warm temperatures dry cooling wet temperature heat moisture
Atom #1775: garden gardens flower flowers vegetable ornamental gardeners gardening nursery floral
Atom #21: dec. nov. oct. feb. jan. aug. 27 28 29 june
Atom #84: celebrations celebration marking festivities occasion ceremonies celebrate celebrated celebrating ceremony
Atom #98: first team start final second next time before test after
Atom #606: vacation lunch hour spend dinner hours time ramadan brief workday
Atom #384: golden moon hemisphere mars twilight millennium dark dome venus magic

! , , , .

, , . , , .

. fastText, RusVectores. 300.

fasttext_model = KeyedVectors.load('/home/astromis/Embeddings/fasttext/model.model')

embeddings = fasttext_model.wv

index2word = embeddings.index2word
embedds = embeddings.vectors

embedds.shape

(164996, 300)

%time
aksvd = ApproximateKSVD(n_components=2000,transform_n_nonzero_coefs=5, )
embedding_trans = embeddings.vectors[:10000]
dictionary = aksvd.fit(embedding_trans).components_
gamma = aksvd.transform(embedding_trans)

CPU times: user 1 µs, sys: 2 µs, total: 3 µs
Wall time: 6.2 µs

dictionary = np.load('./data/mats/dictionary_rus_fasttext_300d.npz')
dictionary = dictionary[dictionary.keys()[0]]

embeddings.similar_by_vector(dictionary[1024,:], 20)

[('', 0.6854609251022339),
 ('', 0.6593252420425415),
 ('', 0.6360634565353394),
 ('', 0.5998549461364746),
 ('', 0.5971367955207825),
 ('', 0.5862340927124023),
 ('', 0.5788886547088623),
 ('', 0.5788123607635498),
 ('', 0.5623885989189148),
 ('', 0.5610565543174744),
 ('', 0.5551878809928894),
 ('', 0.551397442817688),
 ('', 0.5356274247169495),
 ('', 0.531707227230072),
 ('', 0.5174376368522644),
 ('', 0.5131562948226929),
 ('', 0.5120065212249756),
 ('', 0.5077806115150452),
 ('', 0.5074601173400879),
 ('', 0.5068254470825195)]

embeddings.similar_by_vector(dictionary[1582,:], 20)

[('', 0.45191124081611633),
 ('', 0.4515378475189209),
 ('', 0.4478364586830139),
 ('', 0.4280813932418823),
 ('', 0.41220104694366455),
 ('', 0.40772825479507446),
 ('', 0.4047147035598755),
 ('', 0.4030646085739136),
 ('', 0.39368513226509094),
 ('', 0.39012178778648376),
 ('', 0.3866344690322876),
 ('', 0.37968817353248596),
 ('', 0.3728911876678467),
 ('', 0.3663109242916107),
 ('', 0.3640827238559723),
 ('', 0.3474290072917938),
 ('', 0.3473641574382782),
 ('', 0.3468908369541168),
 ('', 0.34586742520332336),
 ('', 0.34555742144584656)]

embeddings.similar_by_vector(dictionary[500,:], 20)

[('', 0.6874514222145081),
 ('-', 0.5172050595283508),
 ('', 0.46720415353775024),
 ('', 0.44713956117630005),
 ('', 0.4144558310508728),
 ('', 0.40545403957366943),
 ('', 0.4030636250972748),
 ('-', 0.4016447067260742),
 ('', 0.38331469893455505),
 ('', 0.37292781472206116),
 ('', 0.3625457286834717),
 ('', 0.35121074318885803),
 ('', 0.3504621088504791),
 ('', 0.34097471833229065),
 ('', 0.33320850133895874),
 ('', 0.3277249336242676),
 ('', 0.3266661763191223),
 ('', 0.31865227222442627),
 ('::', 0.30150306224823),
 ('', 0.2975207567214966)]

itie = index2word.index('')
ispring = index2word.index('')

tie_emb = embedds[itie]
string_emb = embedds[ispring]

simlist = []

for i, vector in enumerate(dictionary):
    simlist.append( (cosine(vector, string_emb), i) )

simlist = sorted(simlist, key=lambda x: x[0])
six_atoms_ind = [ins[1] for ins in simlist[:10]]

for atoms_idx in six_atoms_ind:
    nearest_words = embeddings.similar_by_vector(dictionary[atoms_idx,:])
    nearest_words = [word[0] for word in nearest_words]
    print("Atom #{}: {}".format(atoms_idx, ' '.join(nearest_words)))

Atom #185:          
Atom #1217:         - 
Atom #1213:          
Atom #1978:          
Atom #1796:          
Atom #839:          
Atom #989:          
Atom #414:          
Atom #1140:       -   
Atom #878:          

simlist = []

for i, vector in enumerate(dictionary):
    simlist.append( (cosine(vector, tie_emb), i) )

simlist = sorted(simlist, key=lambda x: x[0])
six_atoms_ind = [ins[1] for ins in simlist[:10]]

for atoms_idx in six_atoms_ind:
    nearest_words = embeddings.similar_by_vector(dictionary[atoms_idx,:])
    nearest_words = [word[0] for word in nearest_words]
    print("Atom #{}: {}".format(atoms_idx, ' '.join(nearest_words)))

Atom #883:          -
Atom #40:          
Atom #215:          
Atom #688:          
Atom #386:          
Atom #676:          
Atom #414:          
Atom #127:          
Atom #592:          
Atom #703:    - -     

#np.savez_compressed('./data/mats/gamma_rus_fasttext_300d.npz', gamma)
#np.savez_compressed('./data/mats/dictionary_rus_fasttext_300d.npz', dictionary)

.

, (Word sense indection), , 1. — , . , , . , , , . , .

UPD: knagaev .