Angular-Based Forgetting Mitigation

• Angular-based Forgetting Mitigation is a continual learning framework that preserves learned feature directions by imposing angular constraints between successive fMRI representations.
• It leverages a teacher-student distillation approach that compares normalized intermediate features, thereby avoiding over-regularization from magnitude differences.
• Empirical results show that angular constraints yield superior retention, significantly reducing performance drops in brain-to-image retrieval tasks compared to traditional methods.

Searching arXiv for the cited papers to ground the article in current preprints. Angular-based Forgetting Mitigation denotes a class of forgetting-control mechanisms that preserve previously acquired behavior by regulating directional change rather than absolute magnitude. In its most specific usage, the term refers to the distillation-style continual learning loss introduced in "BRAIN: Bias-Mitigation Continual Learning Approach to Vision-Brain Understanding," where intermediate fMRI representations from successive training steps are constrained through an angular divergence so that session-to-session adaptation does not overwrite earlier visual-brain correspondences under non-stationary brain signals (Nguyen et al., 25 Aug 2025). Related work uses the same geometric intuition in broader settings, including orthogonality-constrained unlearning updates in parameter space (Zhou et al., 21 Nov 2025) and cosine-based candidate selection in agent-memory control planes, while also showing that angular criteria alone are insufficient for precise deletion semantics (Yang, 14 Jun 2026).

1. Conceptual role in continual vision-brain understanding

Within the BRAIN framework, Angular-based Forgetting Mitigation (AFM) appears alongside De-bias Contrastive Learning (DCL) as one of two new losses. DCL addresses session-wise bias in fMRI signals, whereas AFM is the mechanism used to implement the continual learning loss $\mathcal{L}_{CL}$ that prevents catastrophic forgetting. The continual objective is formulated as

$$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$

where $\mathcal{L}_C$ is the contrastive alignment between fMRI features and visual centroids, $\mathcal{L}_{CL}$ is the forgetting-mitigation term, and $\lambda_{CL}$ controls its influence during optimization (Nguyen et al., 25 Aug 2025).

The motivation is explicitly tied to non-stationary fMRI caused by human memory decay. Later recording sessions are described as weaker, noisier, and less consistent; participants are less certain about stimuli; and the distribution shift is summarized as $p^t(x) \neq p^{t-1}(x)$. In a naïve continual learning setup, gradient-based updates on later-session data can overwrite knowledge acquired from earlier, more reliable sessions. The BRAIN formulation therefore treats each session $t$ as a continual step with data $\mathcal{D}^t$, uses DCL to determine what to learn from biased data, and uses AFM to determine how representations may evolve without losing prior knowledge. In the paper’s own division of labor, DCL is the weighted contrastive alignment mechanism, while AFM is the angular distillation mechanism that preserves representational consistency across steps (Nguyen et al., 25 Aug 2025).

The paper also positions AFM against weight-regularization approaches such as EWC-style constraints. Its argument is that direct parameter constraints are too rigid under session drift and noisy later data, whereas distillation-based constraints on representations are more flexible. AFM therefore operates as a knowledge-distillation regularizer on features rather than as a direct penalty on parameter movement (Nguyen et al., 25 Aug 2025).

2. Mathematical formulation and angular divergence

The continual learning loss is first defined generically over intermediate encoder features:

$$\mathcal{L}_{CL}(x^t) = \frac{1}{L}\sum_{i=1}^L \mathcal{D}_f\big(F_i(x^t,\theta_{t-1}), F_i(x^t,\theta_t)\big) = \frac{1}{L}\sum_{i=1}^L \mathcal{D}_f(z_i^{t-1}, z_i^t),$$

where $F$ is the fMRI encoder, $$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$0 and $$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$1 are the previous and current parameters, $$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$2 denotes the $$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$3-th intermediate layer, and $$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$4 and $$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$5 are teacher and student features from the same input (Nguyen et al., 25 Aug 2025).

AFM is the specific choice of $$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$6 as an angular divergence. The paper contrasts this with the Euclidean alternative

$$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$7

and argues that $$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$8 penalties are sensitive to absolute feature scale. Because feature magnitudes may change as the model adapts to biased and noisy sessions, Euclidean matching can over-regularize and obstruct useful adaptation (Nguyen et al., 25 Aug 2025).

The AFM divergence is then defined as

$$\theta_t^* = \operatorname*{argmin}\ \mathbb{E}_{x^t, y^t \in \mathcal{D}^t}\big[\mathcal{L}_{C}(x^t, y^t) + \lambda_{CL}\mathcal{L}_{CL}(x^t)\big],$$9

Both features are normalized, so the loss becomes insensitive to magnitude and instead measures directional discrepancy. Under the dot-product interpretation highlighted in the paper, the normalized product is $\mathcal{L}_C$0, making the loss approximately $\mathcal{L}_C$1 and encouraging $\mathcal{L}_C$2. The paper notes that an element-wise interpretation would still behave as a direction-similarity metric because the vectors are normalized, but states that the dot-product interpretation is the one that aligns explicitly with the “angular” reasoning and the reference to ArcFace (Nguyen et al., 25 Aug 2025).

With this choice, the AFM loss can be written as

$\mathcal{L}_C$3

The final BRAIN objective combines DCL and AFM as

$\mathcal{L}_C$4

where $\mathcal{L}_C$5 is the final-layer fMRI feature, $\mathcal{L}_C$6 is the visual centroid from a frozen CLIP-based vision encoder, and $\mathcal{L}_C$7 is the session-wise response accuracy or, in an alternative formulation, normalized brain activity (Nguyen et al., 25 Aug 2025).

3. Representation-level operation and training procedure

AFM operates on feature representations rather than on weights or gradients. For each current-step sample $\mathcal{L}_C$8, the previous encoder $\mathcal{L}_C$9 produces teacher features $\mathcal{L}_{CL}$0, the current encoder $\mathcal{L}_{CL}$1 produces student features $\mathcal{L}_{CL}$2, and the angular divergence is computed layer-wise and averaged. The paper states explicitly that the distillation loss uses the features of $\mathcal{L}_{CL}$3 intermediate layers from the fMRI encoder, making AFM a multi-layer feature-distillation loss across encoder depth (Nguyen et al., 25 Aug 2025).

The continual protocol is session-based. The dataset is split as $\mathcal{L}_{CL}$4, each $\mathcal{L}_{CL}$5 corresponding to the fMRI/image pairs collected at step $\mathcal{L}_{CL}$6. At each step, $\mathcal{L}_{CL}$7 is loaded as initialization, the current step is trained on $\mathcal{L}_{CL}$8, and every batch computes both the contrastive loss and the AFM loss. The teacher is used only for forward passes inside AFM and remains frozen, which gives the method the standard teacher-student structure of distillation while keeping the constraint at the representation level rather than the parameter level (Nguyen et al., 25 Aug 2025).

The reported training configuration is AdamW with learning rate $\mathcal{L}_{CL}$9 and cosine decay, batch size $\lambda_{CL}$0, and $\lambda_{CL}$1 epochs per step. The weight on forgetting mitigation is fixed as $\lambda_{CL}$2 across all experiments. AFM itself has no bias weight, no angular margin, and no threshold; it is described as a simple squared angular deviation term. The bias factor $\lambda_{CL}$3 belongs to DCL, not to AFM, although both act on the same inputs during optimization (Nguyen et al., 25 Aug 2025).

The paper also links AFM’s geometry to downstream evaluation. Brain$\lambda_{CL}$4Image retrieval and Image$\lambda_{CL}$5Brain retrieval use cosine similarity, and the authors explicitly argue that preserving feature directions is therefore better aligned with the tasks than penalizing feature magnitudes. This makes AFM a representational geometry constraint that is simultaneously a continual-learning device and a task-aligned similarity regularizer (Nguyen et al., 25 Aug 2025).

4. Empirical behavior and comparative performance

The continual learning experiments compare four settings: W/o CL, LwF, PLOP, and Ours, where “Ours” corresponds to DCL plus AFM. Table 1 reports average continual learning performance for Brain$\lambda_{CL}$6Image and Image$\lambda_{CL}$7Brain retrieval under setups $\lambda_{CL}$8, $\lambda_{CL}$9, $p^t(x) \neq p^{t-1}(x)$0, and $p^t(x) \neq p^{t-1}(x)$1, and Figure 1 shows per-subject trajectories. The qualitative pattern is consistent across these evaluations: W/o CL shows sharp performance decline over steps, LwF and PLOP alleviate forgetting but still degrade over time, and the AFM-based system achieves higher accuracies and smaller drops across setups and subjects (Nguyen et al., 25 Aug 2025).

A concrete example is reported for subject 01 under setting $p^t(x) \neq p^{t-1}(x)$2 for Brain$p^t(x) \neq p^{t-1}(x)$3Image retrieval. W/o CL goes from $p^t(x) \neq p^{t-1}(x)$4 at step 1–20 to $p^t(x) \neq p^{t-1}(x)$5 at step 39–40, a drop of approximately $p^t(x) \neq p^{t-1}(x)$6 points. Under the AFM-based system, the initial performance is the same $p^t(x) \neq p^{t-1}(x)$7, but the final performance is $p^t(x) \neq p^{t-1}(x)$8, a drop of approximately $p^t(x) \neq p^{t-1}(x)$9 points. This is presented as a direct illustration of forgetting mitigation relative both to no continual-learning mechanism and to prior continual-learning baselines (Nguyen et al., 25 Aug 2025).

The ablation study in Table 2 isolates the effect of AFM against $t$0-based distillation. Exp-3 uses DCL + RA with $t$1 forgetting, while Exp-6 uses DCL + RA with AFM. The paper states that Exp-6 achieves $t$2–$t$3 higher performance than Exp-3 across the same subjects, continual learning setups, and retrieval tasks. The comparison is used to support the claim that angular constraints provide better forgetting mitigation and robustness than Euclidean feature matching in this setting (Nguyen et al., 25 Aug 2025).

The same table also compares rehearsal-based and rehearsal-free variants. Exp-5 is a rehearsal-based approach with a $t$4 buffer, but AFM is turned off in that variant, and the paper reports that it performs significantly worse than the rehearsal-free Exp-6. The authors therefore conclude that rehearsal is not an ideal solution for continual vision-brain understanding. They further note that the full continual setup with DCL + AFM surpasses the Non-CL setup that trains on all sessions together, describing this as an uncommon outcome in other domains. The abstract summarizes the aggregate result more broadly by stating that the approach achieves State-of-the-Art performance across various benchmarks, surpassing prior and non-continual learning methods (Nguyen et al., 25 Aug 2025).

5. Broader geometric interpretations beyond BRAIN

A broader formulation of angular forgetting mitigation appears in "Geometric-Disentanglement Unlearning," which treats forgetting side effects as a geometric conflict between forget gradients and the retain-gradient subspace. In that framework, updates are retain-invariant to first order if and only if they are orthogonal, under an optimizer-induced SPD metric $t$5, to the entire subspace spanned by retain gradients. The forget gradient is decomposed into tangential and normal components relative to that subspace, and only the normal component is executed. The resulting Geometric-disentanglement Unlearning (GU) is therefore an angular-based mitigation mechanism in parameter space: side effects correspond to non-zero projection onto protected directions, and mitigation is achieved by projection onto the $t$6-orthogonal complement (Zhou et al., 21 Nov 2025).

GU makes this geometry explicit with projectors

$t$7

where the columns of $t$8 form an $t$9-orthonormal basis of the retain-gradient subspace. The forget update is replaced by $\mathcal{D}^t$0, and under a trust-region constraint the paper proves that this projected direction is the steepest retain-safe descent direction for the forget loss. This use of angle is more stringent than AFM in BRAIN: AFM aligns successive feature directions, whereas GU enforces orthogonality between forgetting updates and retain gradients (Zhou et al., 21 Nov 2025).

A different but related perspective is provided by "Control-Plane Placement Shapes Forgetting," which discusses embedding-space cosine similarity in agent-memory systems. There, an angular rule can be written as a candidate set

$\mathcal{D}^t$1

so that cosine similarity localizes potential memories to mutate. The paper shows that such angular neighborhoods are strong for canonicalization, identifier obfuscation, and cross-lingual matching, but also shows that raw angular thresholds are too blunt for final deletion semantics. Prefix collisions, compound facts, shared attributes, and negation traps require a mutation-time LLM hook that reasons over the candidate set and returns a deterministic edit or deletion plan (Yang, 14 Jun 2026).

Taken together, these works indicate that “angular-based forgetting mitigation” is not a single algorithmic template. In BRAIN, the angle is between old and new intermediate features; in GU, it is between forget updates and retain-gradient subspaces; in agent-memory control planes, it is between query and stored-memory embeddings. This suggests that the term is best understood as a geometric design principle whose concrete meaning depends on whether the protected object is a representation, a gradient subspace, or a memory neighborhood.

6. Assumptions, limitations, and interpretive boundaries

AFM in BRAIN assumes that the direction of feature vectors carries the essential semantics of brain-to-image alignment, while magnitude is more sensitive to noise or scaling. This fits cosine-based retrieval and CLIP-style embeddings, but it also presupposes that the learned fMRI feature space is one in which cosine similarity remains meaningful across sessions. If later sessions induce severe directional drift, the old feature directions may no longer be optimal, and AFM may slow adaptation because it continues to preserve them (Nguyen et al., 25 Aug 2025).

The paper identifies several further limitations. AFM introduces no angular margin or task-aware threshold, unlike ArcFace-style formulations; it relies on the previous model as teacher, so any bias or suboptimality in the teacher can be propagated; and it fixes the number of intermediate layers at $\mathcal{D}^t$2 without exploring which depths matter most. The paper does not provide AFM-specific failure cases, but it does state that these issues can be inferred from the formulation and training setup (Nguyen et al., 25 Aug 2025).

Several misconceptions are also clarified by comparison with adjacent work. Angular mitigation is not identical to weight regularization: AFM preserves representational directions, while EWC-style methods constrain parameter updates directly (Nguyen et al., 25 Aug 2025). Nor is every forgetting-based method angular. "The Effectiveness of Random Forgetting for Robust Generalization" introduces FOMO, which periodically reinitializes a subset of later-layer weights, uses a stable EMA model, and applies consistency regularization to mitigate robust overfitting, but the paper explicitly states that it does not introduce angular or geometric metrics in its forgetting mechanism; any angular interpretation is retrospective rather than procedural (Ramkumar et al., 2024). Likewise, the control-plane study shows that cosine similarity is effective for candidate selection but should not be treated as a sufficient deletion rule in adversarial settings (Yang, 14 Jun 2026).

The BRAIN paper itself suggests several extensions consistent with its geometric premise: angular margin and task-aware AFM, adaptive $\mathcal{D}^t$3, selective AFM on important features, and hybrid angular-plus-magnitude metrics. A plausible implication is that future work may differentiate between directions that encode stable semantics and directions that reflect session-specific bias, rather than preserving all directions uniformly. In that sense, AFM marks a shift from magnitude-sensitive retention toward explicitly geometry-aware retention, but it does not close the broader question of when directional stability should be preserved and when it should be revised (Nguyen et al., 25 Aug 2025).