Debiasing word embeddings¶

This page is the lookup reference for bias geometry in a given static embedding: how gender bias lives in a direction of the vector space, how Bolukbasi et al.'s hard debiasing removes it (identify the subspace, neutralize, equalize), and how WEAT (Caliskan, Bryson & Narayanan, 2017) measures the association that remains — used here to quantify bias before and after debiasing. It states the equations and rules and makes no argument for them; the reasoning behind the mechanics catalogued here — why bias is a direction, why removing it is "Lipstick on a Pig", and where geometric methods give way to learning-time ones — is in this topic's explanation view, reachable from the Crosswalk.

Notation departs from the head word embeddings page deliberately: this topic stores the embedding matrix in the $(V \times d)$ row convention — a word's vector is E[i], the transpose of the head's column layout — because projections, WEAT, and neighbour structure are all transpose-free on rows. What a word vector is and how it is learned live on the head and the word2vec-and-glove pages; this topic treats the vectors as given — a small, hand-planted toy matrix (see Fixture and tolerances) whose every answer is exact and checkable by eye. Cosine similarity and nearest-neighbour ranking are imported from the head rather than restated; this module studies the bias geometry on top of them.

Contents¶

Notation and shapes
The $(V \times d)$ layout and the head relationship
The gender direction
Neutralize
Equalize
Hard debiasing: the pipeline
WEAT
The surviving-bias metric
Public surface
Fixture and tolerances
Failure modes
Implementation Examples
Key Takeaways
References
Crosswalk

Notation and shapes¶

Symbol	Meaning
$V$	vocabulary size: the number of distinct words ($V = 14$ in the fixture)
$d$	embedding dimension: the number of features per word ($d = 4$ in the fixture)
$E$	the embedding matrix, shape $(V \times d)$ — one word per row (this topic's convention)
$i$	a word's integer index, $0 \le i < V$
$e_w$	the embedding of word $w$: a $(d,)$ vector, the row $E[i]$
$B$	a generalized gender subspace (Bolukbasi §5); a single direction `g` is the rank-1 case ($k = 1$)
`vocab`	the word list, sorted, which fixes the row order of $E$ and the `word_to_ix` index map

Every row of $E$ is unit-norm (make_fixture L2-normalizes; see Fixture and tolerances). Unit-norm is required by equalize (its $\sqrt{1 - \lVert \nu \rVert^2}$ term is real only on the unit sphere) and it keeps cosines comparable across the operations. The matrix stores float64.

The $(V \times d)$ layout and the head relationship¶

The head writes the embedding matrix as $E_{\text{head}}$ of shape $(d \times V)$ — one word per column — and a word's vector as the column $E_{\text{head}, :\,i}$. To stay transpose-free on projections, WEAT, and neighbour ranking, this topic takes the transpose and works row-wise:

\[E \;=\; E_{\text{head}}^{\top} \;\in\; \mathbb{R}^{V \times d}, \qquad e_w \;=\; E[i] \;\in\; \mathbb{R}^d.\]

This page	Head (`word-embeddings.md`)
$E$, shape $(V \times d)$	$E_{\text{head}}$, shape $(d \times V)$, with $E = E_{\text{head}}^{\top}$
$e_w = E[i]$ (a row)	$e_w = E_{\text{head}, :\,i}$ (a column)
`E[i]`	`E_head[:, i]`

Where the head's column-convention helpers are reused (cosine similarity and nearest neighbours), the row matrix is passed transposed (its columns are then the rows of this page's $E$). The dossier settles this convention because every operation in this topic — projection on $g$, the neutralize formula, the WEAT s and its pooled standard deviation, and the neighbour ranking — is a row-level computation.

The gender direction¶

Given definitional pairs (he, she), (man, woman), (boy, girl), the gender direction g is the top principal component ($k = 1$) of the centered pair differences (Bolukbasi et al., 2016, §5.1). The procedure:

For each pair $(w_{\text{male}}, w_{\text{female}})$, take the pair mean $\mu = (e_{\text{male}} + e_{\text{female}}) / 2$ and the two deviations $e_{\text{male}} - \mu$ and $e_{\text{female}} - \mu$.
Stack every deviation into one deviation matrix (shape $(2n_{\text{pairs}}, d)$).
Take the SVD; the first right singular vector is the unit gender direction $g$.

The SVD fixes each principal direction only up to sign, so $g$ is pinned male-positive by convention: flip $g$ when $g \cdot (e_{\text{he}} - e_{\text{she}}) < 0$ for the first definitional pair — the same determinism-by-convention the head's pca_project_2d uses. Without the sign rule, two runs could report mirrored projections and a WEAT statistic with a flipped sign.

The method generalizes to a $k$-dimensional subspace $B$ by taking the top $k$ right singular vectors instead of one; the projection onto $B$ becomes $v_B = \sum_{j=1}^{k} (v \cdot b_j)\,b_j$. The fixture's deviation matrix is exactly rank-1 — singular values $[\sqrt{6}, 0, 0, 0]$ — so one direction is the whole subspace here, and every function takes a single unit vector $g$.

gender_direction(E, definitional_pairs) raises ValueError when definitional_pairs is empty.

Neutralize¶

For a gender-neutral word $w$, remove the $g$ component and renormalize:

\[w' \;=\; \frac{w \;-\; (w \cdot g)\,g}{\bigl\lVert w \;-\; (w \cdot g)\,g \bigr\rVert}.\]

After this, $w' \cdot g = 0$ (the direct projection on the gender axis is zero) and $w'$ is back on the unit sphere, so cosines and distances stay comparable.

Zero-residual guard. A word that lies entirely along $g$ has no neutral part: $w - (w \cdot g)\,g$ is the zero vector and renormalizing would divide by zero. neutralize returns the zero vector in that case, which the head's cosine zero-guard then keeps out of every ranking. The fixture's definitional words (he, she, …) are exactly such pure-gender vectors, and hard_debias is structured never to call neutralize on them — those words defined $g$ and are left untouched, as in Bolukbasi.

Equalize¶

For an equality set $E$ (a gendered pair to keep distinct, e.g. {grandfather, grandmother}), Bolukbasi's Step 2a (arXiv:1607.06520) replaces every member's neutral part with the set's shared neutral part $\nu$ and gives the members equal-and-opposite $g$ components. With

\[\mu \;=\; \text{mean}(E), \qquad \mu_B \;=\; (\mu \cdot g)\,g, \qquad \nu \;=\; \mu - \mu_B, \qquad w_B \;=\; (w \cdot g)\,g,\]

each member $w \in E$ becomes

\[w' \;=\; \nu \;+\; \sqrt{1 - \lVert \nu \rVert^2} \;\cdot\; \frac{w_B - \mu_B}{\lVert w_B - \mu_B \rVert}.\]

Every output is unit-norm, the members' $g$ projections are $\pm$ the shared $\sqrt{1 - \lVert \nu \rVert^2}$, and every member is equidistant from any neutralized word — which is what makes the pair unbiased without merging the two words into one vector.

Property	Value
Inputs	the set's member rows, all unit-norm (normalize-first)
Outputs	unit-norm; shared neutral part $\nu$; equal-and-opposite $g$ components
Required condition	$\lVert \nu \rVert \le 1$ (otherwise $\sqrt{1 - \lVert \nu \rVert^2}$ is not real)
Failure mode	`ValueError` when $\lVert \nu \rVert > 1$ (non-unit inputs) or when a member's $g$ component equals the set mean's and the direction $w_B - \mu_B$ is zero

Hard debiasing: the pipeline¶

hard_debias(E, g, neutral_words, equality_sets) is the one-line pipeline that applies the two operations above:

Copy $E$.
For every row index in neutral_words (the gender-neutral occupations), set $E'[i] = \text{neutralize}(E'[i], g)$. The direct bias — the projection $E'[i] \cdot g$ — becomes exactly $0$.
For every equality set in equality_sets, set $E'[\text{set}] = \text{equalize}(E[\text{set}], g)$. The set's $g$ components become $\pm \sqrt{1 - \lVert \nu \rVert^2}$ symmetrically about $g$.

The definitional words that defined $g$ are left untouched, as in Bolukbasi: neutralizing he = +e_0 would annihilate it. The two role lists must be disjoint; hard_debias raises ValueError on overlap rather than return a result whose order-of-operations would silently decide.

WEAT¶

The Word Embedding Association Test (Caliskan, Bryson & Narayanan, 2017) measures the differential association between two attribute sets $A$, $B$ and two target sets $X$, $Y$. For an attribute set this topic uses the male / female definitional words; for a target set, male-coded vs female-coded occupations.

Association score¶

For a single word $w$ and attribute sets $A$, $B$:

\[s(w, A, B) \;=\; \text{mean}_{a \in A}\,\cos(w,\,a) \;-\; \text{mean}_{b \in B}\,\cos(w,\,b).\]

Positive means "leans $A$", negative "leans $B$", zero "unassociated". Cosines come from the head's cosine_similarity (zero-vector guard included).

Test statistic¶

The total association gap between the target sets:

\[\text{statistic} \;=\; \sum_{x \in X} s(x) \;-\; \sum_{y \in Y} s(y).\]

Large positive means the $X$ targets lean toward $A$ more than the $Y$ targets do.

Effect size `d`¶

The gap expressed in pooled standard deviations of the per-target association scores:

\[d \;=\; \frac{\text{mean}_{x \in X} s(x) \;-\; \text{mean}_{y \in Y} s(y)}{\text{std}_{w \in X \cup Y}\, s(w)}.\]

The standard deviation is the population standard deviation (ddof = 0) of the pooled $s$-values; under ddof = 1 the fixture's validated answer would be $1.8102397229$ instead of $1.9830182615$. $d$ is scale-free: it says how far apart the two target groups sit relative to their own spread.

Zero-spread guard. When every pooled $s$-value is equal (no weighted scatter), the std is zero and $d = 0/0$; weat_effect_size returns $0.0$ by definition rather than a nan. This is exactly the post-debias case: every neutralized word scores $0$ against the pure-gender attributes, so the std is $0$, and the dossier pins $d_{\text{after}} = 0$ to this guard.

Exact permutation `p`¶

weat_pvalue(X, Y, A, B) pools the targets $X \cup Y$ and enumerates every $\binom{|X \cup Y|}{|X|}$ equal-size partition $(X', Y')$, computing the statistic of each. The one-sided p-value is the fraction of partitions whose statistic is $\ge$ the observed one — the probability that a random equal-size split associates the targets with the attributes at least as strongly as the true split does. The observed partition always counts itself, so $p \ge 1 / \binom{n}{k}$.

weat_pvalue raises ValueError when $|X| \ne |Y|$ (the test redistributes the pooled targets into equal-size partitions). Exact enumeration replaces Caliskan et al.'s 10,000 sampled permutations; it is affordable here because the fixture's $3$-vs-$3$ target sets give only $\binom{6}{3} = 20$ partitions.

The surviving-bias metric¶

Tested half: for each gender-neutral word, its nearest neighbour within the neutral set — cosine, self excluded, ties broken by ascending index (reusing the head's nearest_neighbors on the neutral sub-matrix alone). The map is {row: neighbour_row}. If horizontal neighbours is identical before and after hard_debias, the words still group by gender through the indirect cluster axis the debiasing never touched — the Gonen–Goldberg climax (Gonen & Goldberg, 2019).

Shown half (the notebook, not this page): a 2-D PCA scatter of the debiased words coloured by original gender (the head's pca_project_2d), in which the two clusters persist though every direct projection is $0$.

Public surface¶

All names are re-exported from dlhub.embeddings (NumPy is always present, so the re-export is eager; make_fixture is reached as dlhub.embeddings.debiasing_word_embeddings.make_fixture so it does not collide with the head's). Full docstrings are on the generated API page; this is the at-a-glance map. Parameters and return shapes match the implementation exactly.

Function	Returns
`make_fixture()`	the planted bundle `{"E", "vocab", "word_to_ix", "dims", "definitional_pairs", "neutral_words", "equality_sets", "weat_sets"}`
`gender_direction(E, definitional_pairs)`	the unit gender direction $g$, shape $(d,)$, male-positive; raises `ValueError` on empty `definitional_pairs`
`neutralize(w, g)`	the neutralized unit vector, shape $(d,)$; the zero vector when $w$ lies entirely along $g$
`equalize(vectors, g)`	the equalized vectors, shape $(n, d)$, in the input's member order; raises `ValueError` when $\lVert \nu \rVert > 1$ or a member's $g$ component equals the set mean's
`hard_debias(E, g, neutral_words, equality_sets)`	the debiased matrix, shape $(V, d)$ — a copy; raises `ValueError` when the two role lists overlap
`weat_association(w, A, B)`	the association score $s(w, A, B)$ (`float`)
`weat_statistic(X, Y, A, B)`	the WEAT test statistic (`float`)
`weat_effect_size(X, Y, A, B)`	the effect size $d$ (`float`); `0.0` when the pooled std is $0$
`weat_pvalue(X, Y, A, B)`	the exact permutation p-value in $[1 / \binom{n}{k}, 1]$; raises `ValueError` on unequal target sizes
`neutral_neighbors(E, neutral_words)`	`dict[int, int]` mapping each neutral word's row to its nearest neutral neighbour's row; raises `ValueError` on fewer than two neutral words

cosine_similarity and nearest_neighbors are imported from dlhub.embeddings.word_embeddings, not reimplemented.

Fixture and tolerances¶

The implementation and its tests share one planted toy embedding set, M = make_fixture(). It is committed as an integer literal, never downloaded or trained, so it cannot drift; every answer is exact and human-checkable by reading the numbers.

Shape: $V = 14$ words by $d = 4$, one word per row (= the head's $E_{\text{head}}^{\top}$). dtype float64.
Coordinates: coordinate 0 is gender (the bias axis — +1 masculine, $-1$ feminine); coordinate 1 is occupation cluster (the indirect-bias axis, correlated with gender — this is what survives neutralize); coordinates 2–3 are within-cluster content (tie-free neighbours).
Vocabulary (sorted, fixing row order and word_to_ix): boy, engineer, girl, grandfather, grandmother, he, man, mechanic, nanny, nurse, programmer, receptionist, she, woman.

Raw rows before make_fixture's row-wise L2 normalization:

Word(s)	Role	Raw $(\text{gender},\ \text{cluster},\ \text{within}_a,\ \text{within}_b)$
`he`, `man`, `boy`	definitional (male)	$(1, 0, 0, 0)$
`she`, `woman`, `girl`	definitional (female)	$(-1, 0, 0, 0)$
`engineer`	male-coded occupation	$(1, 3, 3, 0)$
`programmer`	male-coded occupation	$(1, 3, 2, 0)$
`mechanic`	male-coded occupation	$(1, 3, 0, 0)$
`nurse`	female-coded occupation	$(-1, -3, 3, 0)$
`receptionist`	female-coded occupation	$(-1, -3, 2, 0)$
`nanny`	female-coded occupation	$(-1, -3, 0, 0)$
`grandfather`	equality set	$(2, 0, 0, 1)$
`grandmother`	equality set	$(-1, 0, 0, 1)$

The definitional six are pure-gender archetypes and coincide within gender (he = man = boy = +e_0, she = woman = girl = -e_0). This is intentional: their only role is to define $g$ and to serve as WEAT's gender attributes, and nothing computes neighbours among them — so the coincidence keeps every known answer exact without affecting any result.

WEAT sets. Attributes $A = \{\text{He},\ \text{man},\ \text{boy}\}$ (male), $B = \{\text{she},\ \text{woman},\ \text{girl}\}$ (female); targets $X = \{\text{engineer},\ \text{programmer},\ \text{mechanic}\}$ (male-coded), $Y = \{\text{nurse},\ \text{receptionist},\ \text{nanny}\}$ (female-coded). $|X| = |Y| = 3$, so the exact permutation test enumerates $\binom{6}{3} = 20$ equal-size partitions.

Known answers (the correctness anchors, validated numerically 2026-10-14):

Gender direction. $g = +e_0$ exactly; the SVD of the definitional-pair deviations has singular values $[\sqrt{6}, 0, 0, 0]$ (rank-1), so a single direction ($k = 1$) captures the subspace.
Direct bias (projection on $g$): $\text{engineer},\ \text{programmer}, \ \text{mechanic} = 1/\sqrt{19},\ 1/\sqrt{14},\ 1/\sqrt{10} \approx 0.22942,\ 0.26726,\ 0.31623$; the female occupations are the exact negatives.
WEAT before debiasing: effect size $d = 1.9830182615$, statistic $3.2516189672$, exact permutation $p = 1/20 = 0.05$ (the true split uniquely maximizes the statistic).
After hard_debias: every projection on $g$ is $0$, $d = 0$ exactly (the zero-spread guard), and $p = 1.0$. Equalize acts only on the equality set and does not change the WEAT over $X$ / $Y$.
Survival (the Gonen–Goldberg climax): the neutral-neighbour map is identical before and after neutralize — engineer ↔ programmer, mechanic → programmer, nurse ↔ receptionist, nanny → receptionist. The gendered grouping is intact though every direct projection is $0$.
Equalize {grandfather, grandmother}: the asymmetric inputs $(2, 0, 0, 1)/\sqrt{5}$ and $(-1, 0, 0, 1)/\sqrt{2}$ become $(\pm 0.81663, 0, 0, 0.57716)$ — unit-norm, symmetric about $g$, and equidistant ($\sqrt{2}$) from any neutralized word.

Tolerances:

Comparison	Tolerance
$g$, projections, $d_{\text{after}}$, $p$, the neighbour map, equalize symmetry and norms	exact or `atol = 1e-12`
$d_{\text{before}}$ and $\text{stat}_{\text{before}}$	`atol = 1e-9`
Equalize output vectors	`atol = 1e-9`

Failure modes¶

Symptom	Cause	Fix
Two runs report mirrored $g$ and a WEAT statistic with a flipped sign	the SVD's per-direction sign was left unfixed	flip $g$ when $g \cdot (e_{\text{he}} - e_{\text{she}}) < 0$ for the first definitional pair
`equalize` raises `ValueError` `\|\|nu\|\| > 1`	the equality set's rows are not unit-norm	L2-normalize every row first (the fixture does this in `make_fixture`); equalize's $\sqrt{1 - \lVert \nu \rVert^2}$ is real only on the unit sphere
`equalize` raises `ValueError` `direction undefined`	a member's $g$ component equals the set mean's, so $w_B - \mu_B = 0$	remove that member from the equality set, or split the set
`neutralize` returns the zero vector	$w$ lies entirely along $g$ (pure-gender word, no neutral part)	expected for definitional words; do not neutralize them — `hard_debias` only neutralizes the gender-neutral role rows
`weat_effect_size` returns `nan`	the pooled std is zero but no guard was in place	the guard returns $0.0$ by definition; this is the post-debias case, not a bug
$d_{\text{before}}$ matches $1.81\ldots$ instead of $1.983\ldots$	the sample std (`ddof = 1`) was used	divide by the population std (`ddof = 0`) — the fixture's known answer is pinned to this
`weat_pvalue` raises `ValueError` `equal-size target sets required`	the target sets have different sizes	the permutation test redistributes the pooled targets into equal-size partitions; use balanced target sets
After `hard_debias` the WEAT `d` is still large	`hard_debias` was not applied to the neutral words, or $g$ was computed from the wrong definitional pairs	pass every neutral word index and the corrected $g$; the WEAT sets are unchanged
Direct bias is $0$ yet the neutral-neighbour map is unchanged	the indirect cluster axis survived the debiasing — Gonen–Goldberg	this is the topic's thesis, not a bug; the notebook visualizes the surviving clusters
An equality-set member came back one vector	`hard_debias` and a neutralizer both targeted it	raise on overlap: the two operations would disagree on such a word, and the order they ran in would silently decide the result
$g$ is a $k > 1$ subspace but every function takes one vector	a $k$-dimensional subspace needs a basis, not one direction	take the top $k$ right singular vectors; generalize the projection to $v_B = \sum_j (v \cdot b_j)\, b_j$

Implementation Examples¶

Complete Implementations:¶

Debiasing word embeddings — from scratch (NumPy) - The bias-direction geometry on top of the head's cosine and nearest-neighbour helpers: the planted $(V \times d)$ fixture with its role labels, the top-PC gender direction with its deterministic male-positive sign rule, the neutralize formula (with the pure-gender zero-residual guard), Bolukbasi's equalize formula exactly as arXiv:1607.06520 §A writes it (with its $\lVert \nu \rVert > 1$ and zero-direction guards), the hard_debias pipeline with its disjoint-role invariant, WEAT's s / statistic / effect size d (population std, ddof = 0, with the zero-spread guard) and the exact enumerated permutation p, and the neutral_neighbors map that proves the indirect bias survives. Checked against the fixture: $g = +e_0$ with the rank-1 singular spectrum, the exact projections, WEAT d / statistic / p before ($1.9830182615$ / $3.2516189672$ / $0.05$) and after-debias projections 0, $d = 0$, $p = 1.0$, the neutral-neighbour map identical before and after, equalize symmetry / unit-norm / equidistance, and the d zero-spread guard. Exposes make_fixture, gender_direction, neutralize, equalize, hard_debias, weat_association, weat_statistic, weat_effect_size, weat_pvalue, and neutral_neighbors; imports cosine_similarity and nearest_neighbors from the head.¶

Key Takeaways¶

The embedding matrix here is $E \in \mathbb{R}^{V \times d}$, one word per row, $E = E_{\text{head}}^{\top}$ — the transpose of the head's $(d \times V)$ column convention; a word's vector is $E[i]$, and every row is unit-norm.
The gender direction $g$ is the top principal component ($k = 1$) of the centered definitional-pair differences, taken via SVD of the stacked deviations, with a deterministic male-positive sign rule so $g \cdot (e_{\text{he}} - e_{\text{she}}) > 0$.
Neutralize is $w' = (w - (w \cdot g)\,g) / \lVert w - (w \cdot g)\,g \rVert$: project the $g$ component out and renormalize; a pure-gender $w$ returns the zero vector (the zero-residual guard) so it never poisons a ranking.
Equalize gives an equality set's members the shared neutral part $\nu$ and equal-and-opposite $g$ components: $w' = \nu + \sqrt{1 - \lVert \nu \rVert^2} \cdot (w_B - \mu_B) / \lVert w_B - \mu_B \rVert$, with $w_B = (w \cdot g)\,g$ — every output unit-norm, all equidistant from any neutralized word (Bolukbasi §A, arXiv:1607.06520).
WEAT reports an association $s(w, A, B) = \text{mean}_A \cos(w, a) - \text{mean}_B \cos(w, b)$; a statistic summing $s$ over $X$ and $Y$; an effect size $d$ in the population std (ddof = 0); and an exact permutation p that enumerates every $\binom{|X \cup Y|}{|X|}$ equal-size partition.
The zero-spread guard returns $d = 0.0$ (not nan) when the pooled std is zero — exactly the post-debias case, where every neutralized word scores $0$ against the pure-gender attributes.
After hard_debias the direct bias vanishes and WEAT says "fixed"; the neutral-neighbour map is unchanged (self excluded, ties by ascending index, within the neutral set) — the surviving indirect cluster axis is the Gonen–Goldberg "Lipstick on a Pig" critique in one map.
One planted float64 fixture ($V = 14$, $d = 4$) fixes every answer exactly: $g = +e_0$, WEAT $d_{\text{before}} = 1.9830182615$ / statistic $3.2516189672$ / $p = 0.05$; $d_{\text{after}} = 0$ and $p = 1.0$; the equalize closed form $(\pm 0.81663, 0, 0, 0.57716)$; exact / atol = 1e-12 for the exact anchors, atol = 1e-9 for $d_{\text{before}}$ / $\text{stat}_{\text{before}}$ and the equalize output vectors.

References¶

Bolukbasi, T., Chang, K.-W., Zou, J., Saligrama, V. & Kalai, A. (2016). Man is to Computer Programmer as Woman is to Homemaker? Debiasing Word Embeddings. NeurIPS.
https://arxiv.org/abs/1607.06520 – Hard debiasing: identify the gender subspace, neutralize the neutral words, equalize the equality sets. The exact equalize formula confirmed against §A before coding.
Caliskan, A., Bryson, J. J. & Narayanan, A. (2017). Semantics derived automatically from language corpora contain human-like biases. Science 356(6334), 183–186. – The Word Embedding Association Test: the association score s(w, A, B), the test statistic, and the effect size d. This page uses the exact-enumeration p in place of their 10,000 sampled permutations.
Gonen, H. & Goldberg, Y. (2019). Lipstick on a Pig: Debiasing Methods Cover up Systematic Gender Biases. NAACL.
https://arxiv.org/abs/1903.03862 – Geometric debiasing is superficial: direct bias vanishes while neighbour structure (and downstream classifiers) still encode the bias. The thesis this topic demonstrates with the neutral-neighbour map.
Ng, A. (2018). Deep Learning Specialization, Course 5 (Sequence Models), Week 2 — Natural Language Processing & Word Embeddings.
https://www.coursera.org/specializations/deep-learning – The family's framing; this topic treats the vectors it produces as given and studies the bias geometry on top of them.

Crosswalk¶

Reader wants to	Go to
Learn it step by step	Debiasing word embeddings (notebook)
Look up an equation or rule	Debiasing word embeddings — this page
Understand why it works	Debiasing word embeddings
Learn by running it	Debiasing word embeddings (notebook)

Symbol	Meaning
\(V\)	vocabulary size: the number of distinct words (\(V = 14\) in the fixture)
\(d\)	embedding dimension: the number of features per word (\(d = 4\) in the fixture)
\(E\)	the embedding matrix, shape \((V \times d)\) — one word per row (this topic's convention)
\(i\)	a word's integer index, \(0 \le i < V\)
\(e_w\)	the embedding of word \(w\): a \((d,)\) vector, the row \(E[i]\)
\(B\)	a generalized gender subspace (Bolukbasi §5); a single direction `g` is the rank-1 case (\(k = 1\))
`vocab`	the word list, sorted, which fixes the row order of \(E\) and the `word_to_ix` index map

This page	Head (`word-embeddings.md`)
\(E\), shape \((V \times d)\)	\(E_{\text{head}}\), shape \((d \times V)\), with \(E = E_{\text{head}}^{\top}\)
\(e_w = E[i]\) (a row)	\(e_w = E_{\text{head}, :\,i}\) (a column)
`E[i]`	`E_head[:, i]`

Word(s)	Role	Raw \((\text{gender},\ \text{cluster},\ \text{within}_a,\ \text{within}_b)\)
`he`, `man`, `boy`	definitional (male)	\((1, 0, 0, 0)\)
`she`, `woman`, `girl`	definitional (female)	\((-1, 0, 0, 0)\)
`engineer`	male-coded occupation	\((1, 3, 3, 0)\)
`programmer`	male-coded occupation	\((1, 3, 2, 0)\)
`mechanic`	male-coded occupation	\((1, 3, 0, 0)\)
`nurse`	female-coded occupation	\((-1, -3, 3, 0)\)
`receptionist`	female-coded occupation	\((-1, -3, 2, 0)\)
`nanny`	female-coded occupation	\((-1, -3, 0, 0)\)
`grandfather`	equality set	\((2, 0, 0, 1)\)
`grandmother`	equality set	\((-1, 0, 0, 1)\)

Comparison	Tolerance
\(g\), projections, \(d_{\text{after}}\), \(p\), the neighbour map, equalize symmetry and norms	exact or `atol = 1e-12`
\(d_{\text{before}}\) and \(\text{stat}_{\text{before}}\)	`atol = 1e-9`
Equalize output vectors	`atol = 1e-9`

Symptom	Cause	Fix
Two runs report mirrored \(g\) and a WEAT statistic with a flipped sign	the SVD's per-direction sign was left unfixed	flip \(g\) when \(g \cdot (e_{\text{he}} - e_{\text{she}}) < 0\) for the first definitional pair
`equalize` raises `ValueError` `\|\|nu\|\| > 1`	the equality set's rows are not unit-norm	L2-normalize every row first (the fixture does this in `make_fixture`); equalize's \(\sqrt{1 - \lVert \nu \rVert^2}\) is real only on the unit sphere
`equalize` raises `ValueError` `direction undefined`	a member's \(g\) component equals the set mean's, so \(w_B - \mu_B = 0\)	remove that member from the equality set, or split the set
`neutralize` returns the zero vector	\(w\) lies entirely along \(g\) (pure-gender word, no neutral part)	expected for definitional words; do not neutralize them — `hard_debias` only neutralizes the gender-neutral role rows
`weat_effect_size` returns `nan`	the pooled std is zero but no guard was in place	the guard returns \(0.0\) by definition; this is the post-debias case, not a bug
\(d_{\text{before}}\) matches \(1.81\ldots\) instead of \(1.983\ldots\)	the sample std (`ddof = 1`) was used	divide by the population std (`ddof = 0`) — the fixture's known answer is pinned to this
`weat_pvalue` raises `ValueError` `equal-size target sets required`	the target sets have different sizes	the permutation test redistributes the pooled targets into equal-size partitions; use balanced target sets
After `hard_debias` the WEAT `d` is still large	`hard_debias` was not applied to the neutral words, or \(g\) was computed from the wrong definitional pairs	pass every neutral word index and the corrected \(g\); the WEAT sets are unchanged
Direct bias is \(0\) yet the neutral-neighbour map is unchanged	the indirect cluster axis survived the debiasing — Gonen–Goldberg	this is the topic's thesis, not a bug; the notebook visualizes the surviving clusters
An equality-set member came back one vector	`hard_debias` and a neutralizer both targeted it	raise on overlap: the two operations would disagree on such a word, and the order they ran in would silently decide the result
$g$ is a \(k > 1\) subspace but every function takes one vector	a \(k\)-dimensional subspace needs a basis, not one direction	take the top \(k\) right singular vectors; generalize the projection to \(v_B = \sum_j (v \cdot b_j)\, b_j\)