Tensor Memory (TMEM) Overview

• Tensor Memory (TMEM) is a computational model that uses high-dimensional tensor embeddings to encode, store, and retrieve data by unifying subsymbolic neural and symbolic representations.
• It leverages tensor decompositions and bidirectional feedback mechanisms to facilitate effective semantic and episodic memory operations in complex neural architectures.
• TMEM is implemented in modern hardware, such as TensorDIMM and NVIDIA Blackwell, significantly boosting performance in large-scale deep learning and inference tasks.

Tensor Memory (TMEM) encompasses a class of computational models and hardware mechanisms for encoding, storing, and retrieving information using high-dimensional tensor structures, with particular emphasis on neural, symbolic, and deep learning systems. TMEM models generally unify subsymbolic (vectorial) and symbolic (index-based) representations, leveraging tensor decompositions, learnable embeddings, and layered network architectures for versatile memory and reasoning. TMEM is also now a hardware primitive in state-of-the-art accelerators, directly underpinning modern large-scale deep learning and inference workloads.

1. TMEM in Neurocognitive and Computational Models

TMEM in neural computation is formalized as a layered architecture for memory and perception, as detailed in the "Tensor Brain" framework. TMEM comprises two principal layers:

• Representation Layer (RL): A high-dimensional, subsymbolic "mental canvas" capturing the current cognitive brain state as an activation vector $\boldsymbol\gamma\in(0,1)^n$. This layer aggregates sensory and recurrent inputs, mediating between perception, memory, and attention modules.
• Index Layer (IL): A symbolic index space of discrete ensembles representing concepts, relations (predicates), and temporal indices. Each index $k$ can be "fired" in a winner-take-all or sample-take-all regime to yield an interpreted symbol (Tresp et al., 19 Sep 2024, Tresp et al., 2020).

Encoding, storage, and retrieval entail coupled upward (sensory–to–symbolic) and downward (symbolic–to–sensory) operations, realized by projections and feedback through learned bidirectional connections parameterized by concept embeddings. Each experience (scene, event) is assigned an episodic index with a corresponding embedding $\mathbf{a}_t$, consolidating its cognitive state for future recall.

2. Mathematical Principles: Embeddings, Tensor Decomposition, and Symbol–Subsymbol Interaction

Central to TMEM is the representation of knowledge via tensor-model embeddings:

• Each symbolic index $k$ (entity, predicate, or time) is matched with an embedding $\mathbf{a}_k\in\mathbb{R}^n$, considered the "DNA" of the concept (Tresp et al., 19 Sep 2024, Tresp et al., 2020). These vectors act as bidirectional synaptic weights linking the IL to the RL.
• In bottom-up computation, the current cognitive state $\boldsymbol\gamma$ produces softmax scores $P(Y=k|\boldsymbol\gamma)$ over indices using

$$P(Y=k|\boldsymbol\gamma) = \frac{\exp\left(a_{0,k}+\sum_{i=1}^n a_{i,k}\gamma_i\right)}{\sum_{k'}\exp\left(a_{0,k'}+\sum_{i=1}^n a_{i,k'}\gamma_i\right)}.$$

• Top-down (decoding or embodiment) activates RL by feeding $\mathbf{a}_k$ back, updating $$\mathbf{q}\leftarrow \alpha \mathbf{q} + \beta \mathbf{a}_k$$ and $$\boldsymbol\gamma\leftarrow \mathrm{sigmoid}(\mathbf{q})$$.
• Semantic and episodic memory are implemented as higher-order tensor contractions:

$$\theta_{s,p,o} = \sum_{r_1,r_2,r_3} a_{e_s,r_1} a_{e_p,r_2} a_{e_o,r_3} G^s(r_1,r_2,r_3)$$

for semantic memory (triples), and similarly for episodic memory with time indices and a fourth-order tensor $G^e$ (Tresp et al., 2017).

• The tensor models support probabilistic retrieval and generalization by virtue of their continuous embeddings and multilinear factorization (Tresp et al., 2020).

3. TMEM in Hardware: Near-Memory and On-Chip Architectures

TMEM has significant hardware instantiations:

• TensorDIMM: A near-memory processing module that couples custom DIMMs (DDR4/DDR5) with local vector ALUs and controllers for in-situ tensor operations. The TensorDIMM architecture provides scalable capacity and bandwidth for embedding lookups and elementwise tensor operations, dramatically improving inference throughput (up to $17.6\times$ speedup versus traditional CPU) (Kwon et al., 2019).
• NVIDIA Blackwell TMEM: Blackwell (B200) GPUs implement TMEM as a physically distinct 256 KB scratchpad per SM tightly coupled to 5th-generation Tensor Cores. TMEM exposes explicit instructions in PTX (tcgen05.*), provides 16 TB/s read and 8 TB/s write bandwidth, and reduces tensor access latency to 420 cycles (versus 1000 cycles for H200's global memory). TMEM is explicitly software-managed; it does not share bandwidth with SMEM/L1 and enables accumulation and multi-stage fusion without off-chip traffic (Jarmusch et al., 1 Dec 2025).

Platform	TMEM Capacity (per module)	Read BW	Write BW	Latency (cycles)	Unique Aspects
TensorDIMM (Kwon et al., 2019)	DDRx DIMM (GB–TB scale)	∼25.6 GB/s	∼25.6 GB/s	∼DRAM cycles	Near-DRAM NMP ALU, remote access
Blackwell SM (Jarmusch et al., 1 Dec 2025)	256 KB (per SM)	16 TB/s	8 TB/s	420	On-chip, per-thread tensor pipeline

TMEM mechanisms are critical for supporting bandwidth- and memory-intensive layers in large models, including embeddings, dense/sparse GEMMs, and multi-stage fusion kernels.

4. Unified Operations: Encoding, Storage, Retrieval, and Embodiment

TMEM operational flow consists of:

• Perception loop: Sensory input mapped to RL $\rightarrow$ inference over IL for symbolic labels $\rightarrow$ feedback from IL embeddings to RL, supporting context enrichment and chaining. Scene parsing and semantic triple extraction $(\text{subject}, \text{predicate}, \text{object})$ occur in this loop, both for perception and for memory retrieval (Tresp et al., 19 Sep 2024, Tresp et al., 2020).
• Episodic memory: Assignment of unique time index $t$ and embedding $\mathbf{a}_t$ for each event; recall by direct activation of episodic index and RL reconstruction.
• Semantic memory: Concept indices activate consolidation of related facts and top-down reinstatement of context.
• Embodiment: Top-down activities not only retrieve symbolic facts but also project multimodal representation back to input-proximal layers—enabling sensory imagination, chaining, and context-aware inference.
• Learning: All embeddings and associated parameters are updated via self-supervised gradient-based algorithms, typically maximizing log-likelihoods of self-generated or observed labels, thereby integrating perception, episodic, and semantic traces into a harmonized tensor embedding space (Tresp et al., 19 Sep 2024).

5. TMEM in Tensor-Power and Sequence Models

Tensor memory is also instantiated in sequence models:

• Tensor-Power Recurrent Models: TMEM equips RNNs with explicit "memory buffers" controlled by the degree $p$ of the tensor recurrence. Increasing $p$ extends the autocorrelation memory of the process, at the cost of stability (unbounded Jacobian and possible gradient explosion). Fractional and learnable $p$ allow a trade-off between long memory and dynamical robustness, outperforming vanilla RNN/LSTM models in long-range forecasting tasks (Qiu et al., 2021).

6. Biological, Cognitive, and Theoretical Context

TMEM's mathematical and architectural principles are directly mapped to neurobiological theories:

• Global Workspace Theory: RL models the global workspace; IL implements the symbolic indexing needed for broadcasting and attention (Tresp et al., 2020, Tresp et al., 19 Sep 2024).
• Hippocampal Memory Indexing: Discrete episodic indices correspond to hippocampal pattern separation, while bidirectional embeddings support pattern completion and consolidation into neocortical semantic memory, consistent with Standard Consolidation and Multiple-Trace Theories (Tresp et al., 2017).
• Semantic Decoder: TMEM's tensor structures implement semantic decoding—mapping subsymbolic cognitive states into explicit symbolic facts and relations.

7. Algorithmic and Application Implications

TMEM models and hardware fundamentally alter the constraints and design of scalable AI systems:

• Modeling Implications: Multi-modal, episodic, and semantic information can be co-encoded in a unified tensor space; TMEM enables efficient declarative queries, explicit generative replay, and contextually rich recall (Tresp et al., 19 Sep 2024, Tresp et al., 2020).
• Hardware/Software Optimization: Algorithmic patterns—such as block-fused kernels, on-chip working sets, and tile dimensions for GEMM/Transformer—are dictated by TMEM's bandwidth and latency properties. Blackwell's TMEM, for example, requires 64×64 tiling and double-buffering to saturate throughput; careful attention to bank-conflict patterns and explicit pipeline management is necessary (Jarmusch et al., 1 Dec 2025).
• Performance Scaling: TMEM hardware provides near-ideal scaling for sparse and memory-bound layers, attaining 80–90% of an ideal in-HBM solution for large-scale deep learning workloads with only moderate area and power overhead (Kwon et al., 2019, Jarmusch et al., 1 Dec 2025).

• "How the (Tensor-) Brain uses Embeddings and Embodiment to Encode Senses and Symbols" (Tresp et al., 19 Sep 2024)
• "The Tensor Memory Hypothesis" (Tresp et al., 2017)
• "The Tensor Brain: Semantic Decoding for Perception and Memory" (Tresp et al., 2020)
• "TensorDIMM: A Practical Near-Memory Processing Architecture for Embeddings and Tensor Operations in Deep Learning" (Kwon et al., 2019)
• "On the Memory Mechanism of Tensor-Power Recurrent Models" (Qiu et al., 2021)
• "Microbenchmarking NVIDIA's Blackwell Architecture: An in-depth Architectural Analysis" (Jarmusch et al., 1 Dec 2025)