Naga: Vedic Encoding for Deep State Space Models

State Space Models (SSMs) are increasingly deployed in the Edge because they offer, at comparable performance, a smaller memory/training/inference footprint, compared to Large Language Models (LLMs). These three advantages are a direct consequence of the time recurrence inherent in the SSMs architecture. Here, we further improve this recurrent architecture by positively answering two previously underexplored, orthogonal questions: (1) Can we reduce SSMs memory-footprint without any performance penalty, by also employing depth recurrence? (2) Can we increase SSMs performance by using a fixed and consistent time-granularity across all tasks? The first question is somewhat unexpected, given that SSMs are already recurrent. However, the orthogonal depth recurrence further decreases SSMs memory footprint. We show that a looped SSM with $k$ parameters adaptively iterated $M$ times, achieves a performance comparable to a standard SSM with $k \cdot L$ independent parameters, where $M \leq L$. The second question is also unexpected given the time-recurrent nature of the SSMs architecture. However, it makes perfect sense for the time-parallel training of SSMs on the entire input sequence. We show that concatenating time steps for lower-dimensional sequence elements, or flattening and re-chunking the joint feature-time dimension for high-dimensional ones, can improve the baseline by enhancing the way information is presented to the model. Our results for both extensions lead to consistent benefits across four representative SSM architectures: LRU, S5, LinOSS, LrcSSM.

Time-series data are often sampled irregularly at high frequencies and exhibit long-range dependencies, which makes long-horizon modelling difficult. Continuous-time models such as neural controlled differential equations (NCDEs) and neural rough differential equations (NRDEs) can handle irregular sampling, but they scale poorly to long sequences. Selective state-space models (SSMs) such as Mamba scale linearly with sequence length, but they provide limited recurrent mixing across hidden dimensions within a single block. We propose LogSig-SSM (Log-Signature Compression for State-Space Models), which first compresses long multivariate time series into a shorter sequence of tokens using multi-scale windowed log-signatures, and then processes these tokens with a selective SSM backbone. LogSig-SSM is scalable and robust to irregular sampling, combining log-signature tokens that capture higher-order cross-channel interactions with a selective SSM that models long-range dependencies. The model also admits a continuous-time interpretation as an NCDE/NRDE-style system driven by a log-signature-based input, in which selectivity induces an input-dependent rescaling of the latent dynamics. Across four benchmarks, namely long-sequence classification on UEA, high-frequency physiological regression on PPG-DaLiA, multivariate weather forecasting, and irregularly sampled clinical prediction on PhysioNet Sepsis, LogSig-SSM outperforms or matches strong SSM and continuous-time baselines while training up to $30\times$ faster and using up to $37\times$ less GPU memory than Mamba on the longest sequences.

Deep State-Space Models (Deep SSMs) have shown promising performance and computational benefits in processing of time-series data with complex temporal dependencies. Spiking neural networks (SNNs) constitute a biologically inspired class of deep SSMs, where computing blocks communicate through spikes, discrete single-bit events rather than continuous activation functions. A particularly important task in time-series processing is forecasting, i.e. predicting future values of a sequence from past observations alone. However, how to best perform forecasting with SNNs is an open question, where the standard autoregressive nature of the data processing with SNNs requires values that are not available during the forecasting phase. In this article, we address this challenge. We explore two closed-loop forecasting strategies, in which the network's own generated values are fed back into the network as if they were observed past values: (i) model-level and (ii) layer-level closed-loop forecasting. We evaluate these approaches on the ETT-small datasets, a standard benchmark for time-series forecasting, under both spike-based and continuous activation functions. We compare the proposed closed-loop approaches against state-of-the-art methods such as SpaceTime, as well as the zero-masking baseline in S4. Both of the proposed approaches perform comparably to or better than zero-masking with S4, and in most cases match SpaceTime. These findings highlight the trade-off between continuous and low-bit quantized activations, and demonstrate that the proposed Deep SSM architectures provide a competitive framework for forecasting.