Debiasing and Projection Framework

• The paper introduces a debiasing and projection framework that removes unfair bias by projecting data onto subspaces orthogonal to protected attributes.
• It leverages spectral methods by transforming the adjacency matrix to yield fair densest subgraph extraction under balance constraints.
• Empirical and theoretical results confirm that projection-based debiasing enhances fairness without impairing the inherent signal of the data.

A debiasing and projection framework is a family of methods that removes information aligned with protected or spurious attributes by mapping data, representations, or operators into a subspace in which those directions are suppressed. In "Principal Fairness: Removing Bias via Projections," the framework is instantiated for graph mining: the adjacency matrix is projected onto the orthogonal complement of a binary label vector, and a spectral method is then applied to the transformed operator in order to recover a fair densest subgraph (Anagnostopoulos et al., 2019). In this formulation, fairness is encoded as zero correlation with the protected attribute, and projection becomes the mechanism that turns a potentially biased spectral signal into a fair one.

1. Problem formulation and fairness criterion

The framework in (Anagnostopoulos et al., 2019) studies a graph $G=(V,E,w)$ whose vertices carry a binary label $\ell:V\to\{-1,1\}$, described as red and blue. The target problem is the fair densest subgraph problem: find a subset $S\subseteq V$ maximizing the density

$D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$

subject to the fairness constraint

$|S\cap Red| = |S\cap Blue|.$

The paper’s motivation is explicitly spectral. The usual principal-eigenvector approach to densest subgraph can correlate strongly with the sensitive label, so a dense set recovered from the top eigenvector may be highly imbalanced. The debiasing step is therefore not an auxiliary post-processing heuristic; it is introduced because the principal spectral direction itself can be unfair.

The balance constraint can be written algebraically by introducing the label vector $f\in\mathbb{R}^n$,

$f_i= \begin{cases} \frac{1}{\sqrt{n}}, & \text{if node } i \text{ is red},\[1mm] -\frac{1}{\sqrt{n}}, & \text{if node } i \text{ is blue}. \end{cases}$

If $x$ is a selection vector, then fairness is equivalent to

$f^\top x = 0.$

This representation is central: it recasts demographic balance as orthogonality to a protected-attribute direction. In the framework’s terminology, the desired solution must live in a subspace uncorrelated with $f$.

2. Fair subspace construction

The defining operator of the framework is the orthogonal projector onto the complement of the fairness direction,

$\ell:V\to\{-1,1\}$0

Applied to the adjacency matrix $\ell:V\to\{-1,1\}$1, it yields the transformed matrix

$\ell:V\to\{-1,1\}$2

This construction removes the component aligned with the sensitive attribute and produces what the paper describes as a “fair” spectral embedding. Vectors in the range of $\ell:V\to\{-1,1\}$3 are orthogonal to $\ell:V\to\{-1,1\}$4, so the first-order direction associated with the protected label is suppressed. The framework is therefore projection-based in a literal linear-algebraic sense: it identifies an unfair direction and removes it by orthogonal projection before optimization.

A key property is that the projection leaves every fair solution unchanged. For any fair indicator vector $\ell:V\to\{-1,1\}$5 satisfying $\ell:V\to\{-1,1\}$6,

$\ell:V\to\{-1,1\}$7

Thus the projection is not altering the objective value of feasible fair sets; it is altering the operator so that unfair directions are no longer spectrally dominant.

The paper characterizes this as a soft fairness constraint. Rather than enforcing exact balance directly inside the spectral relaxation, it debiases the representation so that fairness is built into the transformed space. It also notes an important terminological point: although the work is presented within a broader projection-based debiasing philosophy, its core construction is an orthogonal projection, not a Johnson–Lindenstrauss-style random projection (Anagnostopoulos et al., 2019).

3. Spectral relaxation and sweep procedures

The unconstrained spectral relaxation for densest subgraph is

$\ell:V\to\{-1,1\}$8

The fair version replaces $\ell:V\to\{-1,1\}$9 by its projected counterpart: $S\subseteq V$0 After relaxing $S\subseteq V$1 from a Boolean vector to a real vector, the optimum is given by the leading eigenvector of

$S\subseteq V$2

Algorithmically, the method is a sweep over that leading eigenvector. The generalized sweep algorithm computes the top eigenvector $S\subseteq V$3 of $S\subseteq V$4, sorts vertices by decreasing $S\subseteq V$5, and then scans prefix sets. For each prefix $S\subseteq V$6, it evaluates density and retains the best set satisfying approximate balance,

$S\subseteq V$7

This sweep procedure plays the same role as in standard spectral graph methods, but the ranking signal comes from the debiased operator. The framework therefore preserves the computational structure of spectral extraction while modifying the geometry on which the extraction is performed.

The paper applies this machinery directly to the fair densest subgraph problem: given a graph $S\subseteq V$8 and a 2-coloring, find a fair subset $S\subseteq V$9 maximizing $D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$0. Under suitable structural assumptions, the projected spectral relaxation is informative for the hidden fair dense set rather than for the protected label itself (Anagnostopoulos et al., 2019).

4. Guarantees, approximation, and hardness

The theoretical analysis in (Anagnostopoulos et al., 2019) has two layers: recovery under planted structure, and worst-case approximation complexity.

For recovery, the central theorem assumes that the graph contains a fair subset $D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$1 whose induced subgraph $D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$2 is nearly regular and dense, while the ambient graph is sufficiently expanding. More concretely, if

• the graph spectrum satisfies $D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$3, where $D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$4,
• the fair planted subgraph $D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$5 induces a $D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$6-regular subgraph,
• and $D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$7,

then the algorithm recovers all but at most

$D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$8

vertices of $D_S(G)=\frac{2\,|E\cap (S\times S)|}{|S|},$9 in polynomial time. The result formalizes the intended use of the projection: it eliminates the protected direction while preserving enough signal for spectral recovery of the fair dense component.

For approximation, the paper shows that if the whole graph is already fair, then a simple repair of the unconstrained densest-subgraph solution yields a 2-approximation,

$|S\cap Red| = |S\cap Blue|.$0

It further proves that fair densest subgraph is at least as hard as densest-at-most-$|S\cap Red| = |S\cap Blue|.$1 subgraph, and that any $|S\cap Red| = |S\cap Blue|.$2-approximation for densest-at-most-$|S\cap Red| = |S\cap Blue|.$3 induces a $|S\cap Red| = |S\cap Blue|.$4-approximation for fair densest subgraph.

The negative result is sharp under the small set expansion hypothesis. Under SSEH, computing a $|S\cap Red| = |S\cap Blue|.$5-approximation is NP-hard. The paper therefore positions the polynomial-time factor-2 guarantee as essentially optimal under that assumption. A common misconception is that projection-based debiasing in this setting is purely heuristic; the complexity analysis shows otherwise. The projection step is embedded in a framework that also has matching approximation and hardness statements (Anagnostopoulos et al., 2019).

5. Empirical profile on graph benchmarks

The empirical study in (Anagnostopoulos et al., 2019) evaluates the projection-based spectral approach against a combinatorial 2-approximation and several sweep-based heuristics on PolBooks and Amazon product co-purchase graphs.

The principal empirical findings are consistent across the reported settings. The projected spectral methods based on

$|S\cap Red| = |S\cap Blue|.$6

consistently outperform their non-projected counterparts. Among the practical variants, the fair projected sweep (FPS) is often the strongest method, returning balanced subgraphs with high density. On Amazon instances, the spectral methods generally outperform the theoretically optimal 2-approximation heuristic in achieved density, even though the latter has the stronger worst-case guarantee. The projected procedures also reduce the rate of unfair outputs, and paired sweep variants always produce fair solutions by construction.

These results matter because they separate two claims that are often conflated. One claim is representational: removing the protected direction should change the geometry of the spectral embedding. The other is operational: that altered geometry should improve the quality of the returned subgraph under a fairness constraint. The experiments support both claims. They suggest that projection is not merely a fairness regularizer imposed after the fact, but a way of making the principal eigenvector informative for a fair solution (Anagnostopoulos et al., 2019).

6. Broader projection-based debiasing landscape

The graph framework of (Anagnostopoulos et al., 2019) belongs to a broader research line in which debiasing is achieved by removing sensitive directions from a representation before standard optimization or prediction. In that sense it is closely related to fair PCA and fair spectral clustering, both of which use projection to suppress protected-attribute signal before downstream analysis.

Across later work, the same logic is applied to different objects rather than to graph adjacency matrices.

Framework	Object transformed	Projection logic
Multi-attribute INLP (Subramanian et al., 2021)	Hidden representation $|S\cap Red| = |S\cap Blue|.$7	Project onto nullspaces of protected classifiers, iteratively removing principal bias components
Conceptor debiasing (Yifei et al., 2022)	Contextualized embeddings in BERT and GPT	Soft projection via $|S\cap Red| = |S\cap Blue|.$8, with AND/OR for combining bias subspaces
Text-embedding debiasing for VLMs (Chuang et al., 2023)	CLIP text embeddings	Closed-form orthogonal and calibrated projection from biased prompts
PRISM / PRISM-mini (Molahasani et al., 11 Jul 2025)	Shared image-text embedding space	Learn a projection $|S\cap Red| = |S\cap Blue|.$9, or use direct orthogonal projection against discovered bias-attribute embeddings
Dual-branch cross-projection debiasing (Zhao et al., 23 Jun 2026)	Target and spurious branch features	Cross null-space projection $f\in\mathbb{R}^n$0, $f\in\mathbb{R}^n$1

Two distinctions are especially important. First, projection-based debiasing is not always a hard deletion of a subspace. Conceptor methods use a soft projection operator rather than complete nulling (Yifei et al., 2022). Second, not every debiasing framework described in projection-like terms uses an explicit projector. MBD for recommender systems describes a contextual normalization step,

$f\in\mathbb{R}^n$2

as a projection-style transformation into a context-relative scale (Li et al., 15 Mar 2026), whereas FineDeb is explicit that it does not formulate a classic projection operator and instead debiases contextual representations through a pairwise loss (Saravanan et al., 2023).

This broader literature suggests a stable conceptual pattern. Projection frameworks are most natural when the protected or spurious factor can be localized in a recoverable direction or subspace; they become more delicate when bias is distributed, nonlinear, or intersectional. That limitation is visible in later work as well: intersectional evaluations show that naive nullspace projection can suffer rank collapse and can be dominated by bias-constrained training under richer group definitions (Subramanian et al., 2021), while recent VLM methods explicitly note the limits of essentially linear projection for highly complex or non-linear biases (Molahasani et al., 11 Jul 2025). Even so, the basic idea introduced in the fair densest-subgraph setting remains influential: identify the unfair direction, project it away, and perform the main optimization in the residual space.