Signature-Memory Models: A Neuro-Symbolic Architecture for Interpretable and Incremental Learning

Author: Michael San Martim
Affiliation: DataSpoc

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Abstract

This paper introduces Signature-Memory Models (SMMs), a neuro-symbolic framework addressing deep learning's opacity and rigidity by decoupling representation, memory, and reasoning. Unlike parametric models compressing knowledge into weights, SMMs externalize knowledge through signatures—compact codes capturing salient micro-patterns via dimensionality reduction. These signatures are stored in a temporally-indexed associative memory (using FAISS) and retrieved for inference via interpretable symbolic rules. We validate SMMs on Iris, Wine, and MNIST datasets, achieving competitive accuracy (96-97% on smaller datasets, 94% on MNIST) with superior interpretability, memory efficiency (7-25× compression vs. kNN), and incremental learning support. We introduce Temporal SMMs (T-SMMs) incorporating temporal indexing, bridging SMMs with World Models for sequential tasks. Statistical validation, ablation studies, and concrete interpretability examples demonstrate SMMs as a step toward scalable, memory-based, explainable AI.

Keywords: Neuro-Symbolic AI, Associative Memory, Interpretable ML, Incremental Learning, XAI, World Models, Temporal Reasoning

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1. Introduction

Deep neural networks dominate AI but suffer from: (1) Opacity—black-box predictions hindering trust, (2) Rigidity—static knowledge in parameters, and (3) Catastrophic forgetting during sequential learning [1]. Symbolic AI offers transparency but struggles with uncertainty and high-dimensional data [2].

We propose Signature-Memory Models (SMMs), externalizing knowledge into explicit memory. Inputs map to compact signatures (bio-inspired codes), stored with labels and timestamps. Predictions retrieve similar signatures via FAISS indexing and apply symbolic reasoning (e.g., majority vote). This encode-store-retrieve-reason cycle, inspired by cortex-hippocampus interactions [4], provides interpretability while maintaining competitive accuracy.

Connection to World Models: SMMs share World Models' [5] philosophy—explicit architectural separation and compact encodings—but focus on interpretable classification vs. temporal prediction. Temporal SMMs (T-SMMs) bridge this gap via temporal indexing.

Contributions:

1. Neuro-symbolic architecture separating representation, storage, and reasoning
2. Three signature function classes (ordering, micro-combination, spectral)
3. T-SMMs with FAISS temporal indexing
4. Formalized World Models connection
5. Comprehensive empirical validation with statistical rigor
6. Complexity analysis and incremental learning demonstration

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2. Methodology

A Signature-Memory Model is: $\mathcal{M} = (f_{\Sigma}, M, I, R)$ where:

• $f_{\Sigma}: \mathbb{R}^d \to \mathbb{R}^k$ is the signature function
• $M = {(\sigma_i, y_i, t_i)}_{i=1}^N$ is associative memory (signature-label-time triplets)
• $I$ is indexing structure (FAISS)
• $R$ is reasoning mechanism

2.1 Signature Functions

Ordering-Based: $f_{\Sigma}^{\text{ord}}(x) = \text{argsort}(x)[:k]$ (captures feature dominance, scale-invariant)

Micro-Combination: $f_{\Sigma}^{\text{micro}}(x) = [\mathbb{I}(x_{i_1} > x_{i_2}), \ldots]$ (binary comparisons, captures non-linear interactions)

Spectral: $f_{\Sigma}^{\text{spec}}(x) = |\text{FFT}(x)|[:k]$ (frequency patterns for time-series)

Combined: $f_{\Sigma}(x) = [f_{\Sigma}^{\text{ord}}(x); f_{\Sigma}^{\text{micro}}(x)]$ yields $k=15-30$ dims vs. $d=4-784$ original.

2.2 Temporal Memory & FAISS Indexing

Memory stores $(\sigma_i, y_i, t_i)$ with timestamps enabling:

• Recency bias: prioritize recent experiences
• Concept drift handling: adapt to non-stationary distributions
• Episodic retrieval: access specific temporal contexts

FAISS Integration [6]: IndexIVFFlat/HNSW for scalable $O(k \log N)$ retrieval vs. $O(Nk)$ naive search.

Temporal Retrieval: $$\text{Retrieve}(q, t_q, M, I, k, \tau) = \text{top-}k{ (\sigma_j, y_j, t_j) \mid \text{dist}(\sigma_j, q), |t_j - t_q| < \tau }$$

2.3 Reasoning with Temporal Weighting

Distance-Weighted: $\hat{y} = \arg\max_{c} \sum_{j} \omega_d(\text{dist}(\sigma_j, q)) \cdot \mathbb{I}(y_j = c)$

Temporal-Distance (T-SMM): $\hat{y} = \arg\max_{c} \sum_{j} \omega_d \cdot \omega_t(t_q - t_j) \cdot \mathbb{I}(y_j = c)$ with $\omega_t(\Delta t) = e^{-\lambda \Delta t}$

Interpretability: Direct traceability to training examples with distances, timestamps, and voting weights.

2.4 Complexity Analysis

Model	Training Space	Inference (indexed)	Memory Efficiency
kNN	$O(Nd)$	$O(d \log N)$	1×
SMM	$O(Nk)$	$O(k \log N)$	$d/k$ (7-25×)
T-SMM	$O(N(k+1))$	$O(k \log N)$	~$d/k$

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3. Experiments

3.1 Setup

Datasets: Iris (150), Wine (178), MNIST subset (10K)
Baselines: kNN, Decision Tree, Random Forest
Configuration: 10 runs, 70/30 train/test split, $k=20-30$ signature dims
FAISS: IndexIVFFlat (MNIST), IndexFlatL2 (smaller datasets)

3.2 Results

Dataset	SMM	k-NN	Decision Tree	RF
Iris	97.8±1.2%	96.7±1.5%	95.6±1.8%	-
Wine	96.3±1.4%	94.4±1.6%	92.6±2.1%	-
MNIST	94.2±0.3%	96.8±0.2%	86.5±0.5%	97.1±0.2%

Statistical Significance (paired t-test vs. kNN):

• Iris: p=0.08 (marginal), Wine: p=0.04 (significant), MNIST: p=0.001 (kNN better)

Key Findings:

• Competitive on smaller datasets with interpretability advantage
• MNIST gap (2.6%) due to compression loss, but 25× less memory
• More stable (lower std) than Decision Trees

3.3 Ablation Study (Wine)

Signature Type	Accuracy	Dims
Ordering only	92.1±1.8%	10
Micro-combo only	93.5±1.6%	15
Spectral only	89.3±2.3%	10
Combined	96.3±1.4%	20

3.4 Interpretability Example

Test Wine: [13.2, 2.8, 2.3, ...] → Predicted: Class 1 ✓

Rank	Train ID	Label	Distance	Temporal Δ	Weight
1	42	Class 1	0.23	5 samples	0.813
2	67	Class 1	0.31	12 samples	0.763
3	89	Class 1	0.35	8 samples	0.741

Explanation: "Classified as Class 1 due to 4/5 nearest neighbors from Class 1, especially #42 (distance 0.23, recent). Signature matched alcohol, color intensity, flavonoid patterns."

3.5 Incremental Learning

Model	Add 20 Samples (ms)	Test Accuracy
SMM	8.2 (+ 3.5 index)	96.1%
kNN	7.9 (+ 4.1 index)	96.0%
DT	342 (retrain)	95.8%

Concept Drift Simulation:

Batch	SMM	kNN	SMM+Temporal
0-20	96.3%	96.0%	96.3%
41-60	94.9%	94.5%	95.7%
81-100	93.6%	93.2%	95.1%

T-SMM with $\lambda=0.01$ maintains accuracy under drift by down-weighting old memories.

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4. Related Work

4.1 World Models Connection

World Models [5] learn compact environment representations (VAE) + temporal dynamics (RNN) for RL planning.

Similarities:

• Explicit architectural separation (representation/memory/reasoning)
• Compact state encodings for efficiency
• Model-based approaches

Differences:

Aspect	World Models	SMMs	T-SMMs
Goal	Planning, prediction	Classification	Both
Temporal	Explicit (RNN)	Absent	Indexing
Memory	Implicit (RNN states)	Explicit (database)	Explicit + time
Learning	Differentiable	Non-parametric	Non-parametric
Interpretability	Moderate	High	High

Positioning: SMMs = "World Models without temporal prediction"; T-SMMs = "retrieval-based vs. learned dynamics"

4.2 Other Related Work

MANNs [9,10]: Differentiable memory, less interpretable
Metric Learning [11]: Learned embeddings, opaque
CBR [12]: Conceptual predecessor with signature-based formalization
LSH [14]: Compression for retrieval, SMMs add symbolic reasoning

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5. Discussion

Strengths:

• Direct interpretability (exact retrieval traces)
• Modularity (independent component improvement)
• Memory efficiency (7-25× compression)
• Incremental learning without retraining
• Temporal awareness (T-SMMs adapt to drift)

Limitations:

• Hand-designed signatures (MNIST gap)
• Scalability beyond millions unproven
• Non-differentiable (limits deep learning integration)
• Cold-start with sparse data

When SMMs Excel:

• Explainability-critical domains (medical, finance, legal)
• Continual learning (robotics, adaptive systems)
• Small-medium data (n < 10K)
• Non-stationary distributions

When SMMs Struggle:

• Perceptual tasks without pre-trained encoders
• Extremely high-dimensional data
• Maximum accuracy priority over interpretability

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6. Future Work

6.1 Learnable Signatures

Lightweight neural $f_{\Sigma}$ (2-3 layers) trained with contrastive loss, combined with hand-crafted components.

6.2 Scalable Retrieval

Validate on ImageNet (1.2M), web-scale text (10M+) with advanced FAISS (IVF-PQ, GPU HNSW).

6.3 Complex Modalities

• Images: ResNet features → signatures
• Time-series: Wavelet signatures for ECG, sensors
• Graphs/Text: Node2Vec, Sentence-BERT → signatures

6.4 Adaptive Memory Management

• Prototype extraction (k-means clustering)
• Forgetting mechanisms (prune rarely accessed)
• Hierarchical memory (short/long-term with consolidation)

6.5 T-SMMs for RL

Memory: Store $(\sigma_t, a_t, \sigma_{t+1}, r_t, t)$ trajectories

Planning: Retrieve similar transitions → build local graph → MCTS over retrieved experiences

Advantages:

• Interpretable (trace to real experiences)
• No model error (use actual transitions)
• Uncertainty quantification via retrieval density

Applications: Gridworld, CartPole, simple Atari with CNN encoders

6.6 Hybrid SMM-World Model

VAE encoder → signatures → FAISS memory ← RNN dynamics → MCTS planning

Combines perception (World Models) + interpretability (SMMs) + planning.

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7. Conclusion

SMMs externalize knowledge into explicit, temporally-indexed memory, achieving:

• 96-97% accuracy (Iris/Wine), 94% (MNIST)
• Direct interpretability via retrieval traces
• 7-25× memory efficiency vs. kNN
• Incremental learning without retraining
• Temporal adaptation via T-SMMs

We formalized the SMM-World Model relationship: complementary architectures bridging interpretable classification (SMMs) and temporal planning (World Models). Future T-SMMs for RL could bring interpretable model-based RL to safety-critical domains.

Vision: SMMs as "reasoning modules" atop neural feature extractors, bridging the neural-symbolic divide for transparent, adaptable AI in consequential settings (healthcare, finance, justice).

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