RARL: Retentive Angular Representation Learning

• RARL is an incremental open set recognition framework that strategically employs fixed ETF prototypes to anchor class representations and curb drift.
• It uses angular geometry to uniformly allocate inactive prototype slots, allowing accurate mapping of future unknown classes as tasks evolve.
• RARL integrates virtual-intrinsic interactive training with rectification strategies to maintain sharp, discriminative decision boundaries across incremental updates.

Retentive Angular Representation Learning (RARL) is a framework for incremental open set recognition (IOSR) in which unknown representations are encouraged to align around inactive prototypes within an angular space constructed under the equiangular tight frame, thereby mitigating excessive representation drift during knowledge updates. In the formulation introduced for IOSR, RARL combines fixed ETF-based class prototypes, a virtual-intrinsic interactive training strategy, and a stratified rectification strategy to preserve discriminative decision boundaries as classes evolve from unknown to known over a stream of tasks (Yang et al., 8 Sep 2025).

1. Problem setting and motivation

RARL is defined for a streaming, task-based setting in which training data arrive as a sequence of tasks,

$\{D_{\text{train}}^1, \dots, D_{\text{train}}^T\},$

with

$$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$

and disjoint label spaces across tasks,

$$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$

At task $t$, the model has access only to the current task data and a small rehearsal buffer $\mathcal{M}$ containing a few exemplars per class from previous tasks. Evaluation uses

$D_{\text{test}}^t = D_K^t \cup D_U^t,$

where known samples have label space

$Y_K^t = \bigcup_{j=1}^t Y_{\text{train}}^j,$

and unknown samples are defined by the next task,

$Y_U^t = Y_{\text{train}}^{t+1}.$

The learning objective is written as

$$\underset{f \in \mathcal{H}}{\arg\min} \left\{ \mathcal{R}_\epsilon(f, D_K^t) + \alpha \cdot \mathcal{R}_o(f, D_U^t) \right\}.$$

This definition is significant because the unknown set is evolving: the unknown classes at step $t$ are precisely the classes that must later become known (Yang et al., 8 Sep 2025).

The framework is motivated by limitations in both static OSR and conventional class-incremental learning. Static OSR assumes a fixed set of known classes and unknowns that never become known, so it does not address how newly discovered unknown classes are incorporated or how previously learned decision boundaries are preserved. Standard CIL, by contrast, is closed-set: it assumes every test instance belongs to one of the known classes, does not explicitly model unknowns or open-space risk, and may push representations away from inactive prototypes in ways that harm unknown detection. RARL is therefore designed for a setting where representation drift, inter-class confusion, and decision-boundary instability accumulate over time.

A central misconception addressed by this formulation is that IOSR unknowns are merely out-of-distribution noise. In RARL, they are future task classes. That distinction alters both the geometry of the feature space and the retention problem: the model must reject unknowns now without destroying the possibility of integrating them later.

2. Angular geometry and fixed ETF prototype space

RARL organizes the representation space entirely in angular terms. Let $$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$0 denote the backbone feature extractor. Features and prototypes are $$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$1-normalized,

$$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$2

and classification scores are cosine similarities $$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$3 scaled by a learnable $$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$4. The key architectural constraint is that the prototypes are fixed to an Equiangular Tight Frame rather than learned. Only prototypes for currently known classes are active in the softmax; the remaining prototypes are inactive and excluded from the denominator (Yang et al., 8 Sep 2025).

The ETF scaffold is defined by a prototype matrix

$$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$5

constructed as

$$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$6

where $$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$7. The pairwise inner products satisfy

$$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$8

This produces equal angles between all class directions, a simplex-like symmetric configuration that maximizes inter-class angular separation and provides balanced margins between all directions.

Classification over intrinsic classes is then expressed as

$$D_{\text{train}}^t = \{(x_i^t, y_i^t)\}_{i=1}^{n_t}, \qquad y_i^t \in Y_{\text{train}}^t,$$9

The important operational detail is that, for a given task, only prototypes corresponding to seen classes are active in the denominator. This “releases space around inactive prototypes”: the loss does not push features away from those reserved directions.

The consequence is twofold. First, prototypes never move, so incremental expansion does not require reconfiguring the geometric scaffold. Second, because inactive ETF vertices remain unused during current-task training, unknown samples can occupy open regions associated with future classes. In the RARL interpretation, retention is therefore achieved not by continually reshaping class weights, but by anchoring semantics to a fixed angular frame.

3. Virtual–intrinsic interactive training and stratified rectification

RARL sharpens boundaries around intrinsic classes through Virtual–Intrinsic Interactive (VII) training. Intrinsic classes are the real task classes mapped to fixed ETF prototypes. Virtual classes are synthetic, boundary-proximal classes constructed within a batch by mixing samples from different intrinsic classes. For each intrinsic sample $$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$0 in a batch with $$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$1 intrinsic classes, the corresponding virtual instance is

$$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$2

Each virtual class has its own learnable prototype $$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$3, distinct from the fixed ETF prototypes. A virtual-class classification loss

$$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$4

clusters virtual features around their own prototypes and prevents them from collapsing into a single ambiguous auxiliary cluster (Yang et al., 8 Sep 2025).

The VII loss explicitly couples intrinsic and virtual samples. Using

$$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$5

the batch loss is

$$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$6

with

$$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$7

This yields three effects stated in the formulation: virtual alignment, virtual discrimination, and intrinsic–virtual separation. Virtual features are pulled toward their own virtual prototypes, pushed away from other virtual prototypes, and intrinsic features are pushed away from their corresponding virtual boundary prototypes. The result is a sharper partition in which intrinsic features move deeper into their class interiors.

Because VII contains one positive term and many negative terms, RARL adds Positive–Negative Boundary Rectification (PNBR) to rebalance the force field. PNBR transforms cosine similarity as

$$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$8

where $$Y_{\text{train}}^{t_1} \cap Y_{\text{train}}^{t_2} = \emptyset,\quad t_1 \neq t_2.$$9 is learnable. In the description of the method, this strengthens positive virtual alignment while softening negative repulsion, thereby preventing virtual features from being pushed too far into open space. Old–New Boundary Rectification (ONBR) addresses a different imbalance: the dominance of new-class samples over the small rehearsal buffer. The method states that ONBR introduces a hyperparameter $t$0 and uses $t$1 instead of $t$2, applying this transformation to old-class cosine similarities so that old classes receive more alignment pressure during incremental updates.

The complete training objective is

$t$3

with

$t$4

Here $t$5 is a less-forget feature distillation term between normalized features from the frozen previous model and the current model. The overall design jointly anchors semantics via ETF, compacts intrinsic and virtual structure, rectifies imbalance-induced distortions, and constrains drift across increments.

4. Retention mechanism, unknown modeling, and incremental update procedure

RARL implements retention through a fixed geometric scaffold plus controlled incremental activation. In the base task, the method constructs the ETF space, marks prototypes of base classes as active, trains with $t$6, $t$7, $t$8 with PNBR, and the less-forget constraint, and then stores a small number of samples into memory $t$9. In subsequent tasks, each new class is assigned to a previously inactive ETF prototype; all previously active prototypes remain fixed. Training uses current task data together with rehearsal samples from $\mathcal{M}$0, the same losses as in the base stage, and ONBR to counteract old-class boundary drift. After training, memory is updated with 20 exemplars per class (Yang et al., 8 Sep 2025).

This procedure makes the retention claim precise. No old data beyond the rehearsal buffer is stored, yet class anchors are preserved because prototypes never move. A newly discovered class is not inserted by reallocating existing directions; it is mapped to an already reserved inactive ETF vertex. This directly addresses the representation-drift problem identified in IOSR.

Unknown modeling in RARL is implicit rather than generative. Known classes occupy active ETF directions that are evenly spread. Inactive prototypes reserve symmetric slots for future classes, and the loss does not push features away from those inactive directions. The description therefore suggests that unknown samples tend to fall into low-confidence angular regions relative to active prototypes, often geometrically closer to inactive vertices or open space.

The paper’s equations focus on training, but the stated inference perspective is standard for cosine-based OSR: a sample is assigned to the known class with highest softmax probability if that score is high enough, otherwise it is classified as unknown. In that interpretation,

$\mathcal{M}$1

and $\mathcal{M}$2 is compared against a threshold. This is not a separate unknown classifier; it is a decision rule induced by the angular structure of the learned feature space.

5. Experimental protocol, benchmark results, and ablations

RARL is evaluated on CIFAR-100 and TinyImageNet under IOSR protocols in which unknown test samples at step $\mathcal{M}$3 are drawn from the next task’s label space. On CIFAR-100, the protocols are Base-20 with 8 steps and Base-20 with 4 steps. On TinyImageNet, the protocols are Base-50 with 10 steps and Base-60 with 7 steps. The implementation uses a ResNet-34 backbone, SGD with learning rate $\mathcal{M}$4, momentum $\mathcal{M}$5, weight decay $\mathcal{M}$6, 160 epochs per task, learning-rate decays at 80 and 120 epochs, and a memory of 20 exemplars per class. Performance is reported with ACC for known-class classification, AUROC for known-versus-unknown discrimination, and OSCR for joint open-set classification behavior (Yang et al., 8 Sep 2025).

Protocol	LUCIR	RARL
CIFAR-100, Base-20/8	Avg ACC 57.18, Last ACC 46.17; Avg AUROC 64.15, Last AUROC 58.02; Avg OSCR 44.55, Last OSCR 34.85	Avg ACC 62.56, Last ACC 48.53; Avg AUROC 66.84, Last AUROC 61.69; Avg OSCR 48.86, Last OSCR 37.37
CIFAR-100, Base-20/4	Avg ACC 66.55, Last ACC 53.86; Avg AUROC 68.51, Last AUROC 60.73; Avg OSCR 52.83, Last OSCR 40.67	Avg ACC 70.63, Last ACC 57.71; Avg AUROC 68.77, Last AUROC 64.93; Avg OSCR 54.91, Last OSCR 44.72
TinyImageNet, Base-50/10	Avg ACC 47.56, Last ACC 36.41; Avg AUROC 64.79, Last AUROC 65.36; Avg OSCR 39.08, Last OSCR 30.24	Avg ACC 52.93, Last ACC 39.94; Avg AUROC 65.46, Last AUROC 66.29; Avg OSCR 42.69, Last OSCR 33.13
TinyImageNet, Base-60/7	Avg ACC 50.96, Last ACC 39.04; Avg AUROC 66.39, Last AUROC 66.25; Avg OSCR 42.33, Last OSCR 32.65	Avg ACC 55.49, Last ACC 42.26; Avg AUROC 67.27, Last AUROC 65.64; Avg OSCR 45.08, Last OSCR 34.48

Across the reported protocols, RARL improves over the strongest cited CIL baseline, LUCIR. On CIFAR-100, the average ACC improvement over LUCIR is $\mathcal{M}$7 to $\mathcal{M}$8, AUROC improves by up to $\mathcal{M}$9, and OSCR improves by $D_{\text{test}}^t = D_K^t \cup D_U^t,$0 to $D_{\text{test}}^t = D_K^t \cup D_U^t,$1. On TinyImageNet, RARL also improves both closed-set and open-set metrics, though the gains are somewhat smaller on AUROC in some settings.

The ablation study on CIFAR-100 Base-20/8 further isolates the components. SoftMax using all fixed prototypes, active and inactive, in the denominator yields the worst result with Avg OSCR $D_{\text{test}}^t = D_K^t \cup D_U^t,$2, confirming that including future or inactive prototypes harms open-set performance. Adding only $D_{\text{test}}^t = D_K^t \cup D_U^t,$3 raises Avg OSCR to $D_{\text{test}}^t = D_K^t \cup D_U^t,$4. Adding $D_{\text{test}}^t = D_K^t \cup D_U^t,$5 gives $D_{\text{test}}^t = D_K^t \cup D_U^t,$6. Adding $D_{\text{test}}^t = D_K^t \cup D_U^t,$7 without PNBR slightly lowers performance to $D_{\text{test}}^t = D_K^t \cup D_U^t,$8, indicating that imbalanced VII can overcompact features and reduce open space. Adding PNBR raises Avg OSCR to $D_{\text{test}}^t = D_K^t \cup D_U^t,$9. ONBR alone, without virtual or VII components, gives $Y_K^t = \bigcup_{j=1}^t Y_{\text{train}}^j,$0. Full RARL reaches the best Avg OSCR of $Y_K^t = \bigcup_{j=1}^t Y_{\text{train}}^j,$1. The reported t-SNE visualizations are described as showing more compact and better-separated clusters than LUCIR while preserving structure for future unknowns.

6. Relation to earlier angular representation learning, limitations, and extensions

An interpretable precursor to RARL’s angular emphasis appears in the speaker-verification framework presented in "Discriminative Speaker Representation via Contrastive Learning with Class-Aware Attention in Angular Space" (Li et al., 2022). That work places $Y_K^t = \bigcup_{j=1}^t Y_{\text{train}}^j,$2-normalized embeddings on a hypersphere, formulates supervised contrastive learning with cosine similarity, introduces an additive angular margin through

$Y_K^t = \bigcup_{j=1}^t Y_{\text{train}}^j,$3

and uses class-aware attention to reduce the harmful influence of extremely hard or mislabeled negatives. It also combines a class-centric angular objective, AAMSoftmax, with an instance-level angular contrastive objective using gradient-based multi-objective optimization. The term “retentive” is not used there, but this suggests a broader angular design pattern in which compact intra-class cones, greater inter-class angular separation, and outlier-robust weighting contribute to stable discriminative geometry (Li et al., 2022).

RARL differs in scope and mechanism. Its primary target is IOSR rather than speaker verification, its prototypes are fixed to an ETF rather than learned class weights, and its notion of retention is explicitly incremental: unknowns are future classes, inactive prototypes are pre-allocated, and geometric stability is preserved through fixed directions, VII-based boundary shaping, rectification, and feature distillation. A plausible implication is that RARL extends angular-margin thinking from static metric learning toward a continual open-world setting.

The method also has explicit limitations. It requires a prior upper bound $Y_K^t = \bigcup_{j=1}^t Y_{\text{train}}^j,$4 on the total number of classes in order to allocate ETF prototypes. Its loss structure is comparatively complex, with ETF classification, virtual classes, VII, PNBR, ONBR, and $Y_K^t = \bigcup_{j=1}^t Y_{\text{train}}^j,$5 interacting through shared angular geometry. Virtual classes add computational and memory overhead. Unknown modeling remains implicit, relying on angular structure and thresholding rather than a separate generative or density model. These are design trade-offs rather than contradictions: the fixed scaffold improves retention, but only under a pre-allocated prototype budget and a carefully balanced optimization scheme (Yang et al., 8 Sep 2025).

The extensions proposed alongside RARL include application to other modalities and domains, combination with task-free or online continual learning methods such as DYSON, stronger theoretical analysis of open-space risk and representation drift in ETF-based IOSR, and adaptive prototype allocation when the actual number of classes diverges from the initial budget. Within that trajectory, RARL can be understood as a specific formulation of retentive angular learning in which continual expansion is handled by reserving, protecting, and later activating angular regions for classes that are initially unknown.