Decomposition-LSTM: Modular Recurrent Models

• Decomposition-LSTM (DLSTM) is a family of architectures that decomposes signals, latent factors, or model parameters to streamline LSTM processing and recomposition.
• DLSTM methods enhance prediction accuracy and interpretability by isolating trends, seasonalities, and noise, as demonstrated in diverse tasks like forecasting and classification.
• Applications range from financial forecasting to solar flare prediction, showing quantitative gains in metrics such as RMSE, AUC, and MSE compared to standard LSTM models.

Decomposition-LSTM (DLSTM) denotes a family of architectures that combine some form of decomposition with long short-term memory (LSTM) sequence modeling. In the literature, the decomposed object is not uniform: it may be the observed time series itself, a tensorized representation of multivariate data, latent temporal factors, the LSTM hidden state, the LSTM parameter matrices, or even the functional graph of a trained network. Correspondingly, the downstream use of the LSTM also varies, including per-component forecasting with recomposition, factor-sequence prediction followed by tensor reconstruction, decomposition-aware classification, interpretable hidden-state attribution, post-hoc modular reuse, and set-based over-approximation of recurrent dynamics (Boadi, 11 Sep 2025, Hassani et al., 7 Jul 2025, Joneidi et al., 2019, Wang et al., 13 May 2026, Charlier et al., 2019, Bilokon et al., 2023, Murua et al., 2020, Wang et al., 2023, Shi et al., 2021, Imtiaz et al., 2022, Glunt et al., 19 Mar 2025).

1. Definition and conceptual scope

The surveyed literature indicates that DLSTM is not a single standardized architecture. In one line of work, DLSTM refers to a pipeline that first decomposes a non-stationary signal into smoother components and then fits one LSTM per component, as in hybrid VMD-LSTM forecasting for Bitcoin prices (Boadi, 11 Sep 2025). In another, DLSTM denotes a model with an internal moving-average decomposition block that splits a sequence into trend-cyclical and seasonal components, discards random noise, and feeds the retained components to an LSTM-based classifier, as in solar flare prediction (Hassani et al., 7 Jul 2025). A closely related construction appears in electronic trading, where a trend series and a remainder series are each processed by separate LSTM branches and fused for one-step-ahead mid-price movement classification (Bilokon et al., 2023).

A broader usage treats decomposition as tensor or factor analysis preceding or restructuring recurrent modeling. Large-scale spectrum occupancy learning uses CP decomposition of a three-way tensor and then applies LSTMs to the temporal factor matrix before reconstructing future slices (Joneidi et al., 2019). Electric-vehicle state-of-charge prediction uses Tucker-style reduction on the feature mode of a sample × window × feature tensor and passes the resulting latent sequences to a stacked LSTM (Wang et al., 13 May 2026). Smart-contract profiling uses non-negative PARATUCK2 to extract time-varying latent factors, whose diagonal trajectories are modeled by LSTM and then mapped back into future interaction tensors (Charlier et al., 2019). A still more internal form of DLSTM factorizes the LSTM weight matrices themselves by tensor trains, replacing dense gate matrices with MPS- or MPO-structured parameterizations (Murua et al., 2020).

Other works shift decomposition from the input signal to the recurrent representation or the trained model. DeLELSTM decomposes a shared hidden state into long-term and instantaneous per-variable contributions using a tensorized LSTM and a linear approximation layer (Wang et al., 2023). Oracle-supervised learning of bounded context-free grammars partitions the LSTM hidden state into subspaces for PDA state and stack positions (Shi et al., 2021). Post-hoc RNN decomposition splits a trained LSTM into reusable or replaceable per-label modules (Imtiaz et al., 2022). Functional decomposition for hybrid-zonotope analysis rewrites the LSTM cell as a directed acyclic graph of observables for set-based over-approximation (Glunt et al., 19 Mar 2025).

2. Canonical decomposition–recurrent–recomposition pattern

The clearest DLSTM pattern is a three-stage workflow: decompose the original sequence, model the decomposed parts with LSTMs, then recompose or fuse the outputs. In the Bitcoin forecasting model, the original normalized closing-price series is decomposed by Variational Mode Decomposition (VMD) into $K=15$ intrinsic mode functions (IMFs); each IMF is passed to its own LSTM with a 30-step lookback window, and the component forecasts are summed as

$\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$

after which inverse MinMax scaling returns the forecast to the dollar scale (Boadi, 11 Sep 2025). The decomposition is explicitly frequency-oriented: IMF 1 captures a long-term trend, IMF 7 medium-frequency oscillations, and IMF 15 high-frequency low-amplitude noise.

In solar flare prediction, the decomposition is an internal model block rather than an external preprocessing stage. The sequence $X$ is split by average pooling into a trend-cyclical component and a seasonal component,

$$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$

while the residual random noise is excluded from the predictive path (Hassani et al., 7 Jul 2025). The resulting trend and seasonal parts are passed through a fully connected layer, one LSTM layer, and two additional fully connected layers for binary classification of small versus large flares. This formulation makes the decomposition itself part of the network architecture.

The trading DLSTM applies the same basic logic to limit-order-book sequences. A moving-average trend series $X_t$ and a remainder series $X_r$ are computed, each is processed by its own LSTM, and the branch outputs $H_t$ and $H_r$ are added before a linear layer and softmax produce the three-class mid-price movement prediction (Bilokon et al., 2023). Unlike multi-step sequence-to-sequence forecasters, this model is explicitly designed for one-step-ahead classification, so decomposition is used to separate low-frequency drift from high-frequency microstructure effects rather than to support long-horizon decoding.

A plausible synthesis is that the most canonical meaning of DLSTM is not “an LSTM with one special cell,” but “a decomposition-aware recurrent architecture” whose essential logic is decompose $\rightarrow$ specialize $\rightarrow$ recombine.

3. Decomposition operators and where they enter the model

A central axis of variation in DLSTM is the type of decomposition operator and the location at which it is applied.

For signal-level decomposition, the operators are time-series transforms. VMD produces band-limited IMFs under a constrained variational formulation and is used to reduce non-stationarity and separate trend, cycles, and noise before per-IMF forecasting (Boadi, 11 Sep 2025). Moving-average or average-pooling decomposition plays a similar role in solar flare prediction and electronic trading, where the split is formulated as trend-cyclical plus seasonal, or trend plus remainder, respectively (Hassani et al., 7 Jul 2025, Bilokon et al., 2023).

For latent-factor decomposition, the operators are tensor factorizations. In spectrum occupancy learning, daily power spectral density measurements are arranged as a tensor $\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$0, decomposed by CP as

$\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$1

and the columns of $\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$2 are forecast by separate LSTMs before the full tensor is reconstructed (Joneidi et al., 2019). In the EV setting, a three-way sample × window × feature tensor is reduced only along the feature mode, effectively applying a Tucker-style projection that maps 14 features to 10 latent components while preserving the time axis for LSTM processing (Wang et al., 13 May 2026). In smart-contract profiling, PARATUCK2 decomposes each slice as

$\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$3

with the time-varying diagonal entries of $\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$4 and $\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$5 extracted as latent trajectories for LSTM forecasting (Charlier et al., 2019).

For parameter-level decomposition, the target is not the data but the recurrent map itself. Tensor-train DLSTMs replace the stacked input-to-hidden and hidden-to-hidden matrices $\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$6 and $\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$7 with MPS or MPO factorizations, so gate computations are implemented via tensor contractions rather than dense matrix multiplications (Murua et al., 2020). This shifts DLSTM toward model compression and edge deployment.

For representation-level decomposition, the hidden state is the object being factorized. DeLELSTM introduces a tensorized LSTM with one hidden vector per variable and approximates the shared hidden state by

$\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$8

thereby separating long-term and instantaneous effects (Wang et al., 2023). Oracle-supervised PDA learning hard-partitions $\hat{y}(t) = \sum_{k=1}^{K} \hat{y}_k(t),$9 into a state slice and stack-position slices, enforcing an explicit latent-space decomposition (Shi et al., 2021).

This diversity shows that “decomposition” in DLSTM is not restricted to denoising or trend extraction. It can act on observations, latent variables, parameters, or symbolic computational structure.

4. Reported applications and quantitative performance

DLSTM methods have been reported in financial forecasting, solar-activity prediction, spectrum occupancy learning, battery management, smart-contract profiling, and electronic trading. The tasks span regression, classification, detection, and sequence reconstruction rather than a single benchmark family.

Domain	DLSTM form	Representative reported result
Bitcoin price forecasting	VMD into 15 IMFs, one LSTM per IMF, summed forecast	Test RMSE $X$0 vs $X$1, MAE $X$2 vs $X$3, $X$4 vs $X$5 for plain LSTM (Boadi, 11 Sep 2025)
Solar flare prediction	AvgPool trend-cyclical/seasonal decomposition, regularization, ensemble	Best regularized ensemble DLSTM reaches AUC $X$6, TSS $X$7, and large-flare recall $X$8–$X$9 (Hassani et al., 7 Jul 2025)
Spectrum occupancy learning	CP decomposition of $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$0 tensor, LSTM on temporal factors	LSTM+CPD achieves error $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$1 and total time $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$2 s versus $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$3 and $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$4 s for plain LSTM (Joneidi et al., 2019)
EV state-of-charge prediction	Tucker-style feature-mode reduction, 2-layer LSTM	MSE $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$5 vs $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$6, RMSE $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$7 vs $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$8, $$X_1 = \mathrm{AvgPool}(\mathrm{Padding}(X)), \qquad X_s = X - X_1,$$9 vs $X_t$0 (Wang et al., 13 May 2026)
Smart contracts / VoD	Non-negative PARATUCK2 temporal factors forecast by LSTM	Smart contracts: average MAE $X_t$1, average MDA $X_t$2; Movielens: MAE $X_t$3, MDA $X_t$4 (Charlier et al., 2019)
Electronic trading	Trend/remainder two-branch DLSTM for movement classification	Accuracy $X_t$5 at horizon 20; with $X_t$6 transaction cost, highest CPR and SR across horizons 20, 30, 50, and 100 (Bilokon et al., 2023)

Across these applications, the reported gains are attributed to several recurring mechanisms: reduction of non-stationarity and complexity in raw financial series, explicit isolation of random noise from signal, preservation of multiway structure in tensors, and compression of redundant feature directions before recurrent modeling. The reported evidence also indicates that decomposition is not inherently tied to one target type: the same design principle is used for 30-day-ahead price levels, rare-event solar classification, occupancy detection, SOC regression, and order-book movement prediction.

5. Interpretability, modularity, and formal analysis

DLSTM has also been developed as an interpretability and analysis framework rather than only a predictive architecture. DeLELSTM uses a standard LSTM together with a tensorized LSTM, then solves a least-squares decomposition of the shared hidden state into historical and fresh-information terms. This yields explicit instantaneous and long-term scores for each variable, including the instantaneous importance

$X_t$7

and the global importance

$X_t$8

which were used to analyze PM2.5, electricity consumption, and exchange-rate data (Wang et al., 2023). In this setting, decomposition turns the hidden state into an additive explanatory object.

A different interpretability use arises in bounded context-free grammar learning. There, oracle training enforces a forced decomposition of the LSTM hidden state into contiguous segments representing PDA control state and bounded stack positions. Under this supervision, both LSTMs and Transformers can simulate bounded PDAs well, but without forced decomposition LSTMs largely fail to capture stack behavior, whereas Transformers are much less affected (Shi et al., 2021). The result is significant because it ties DLSTM-style decomposition to representational capacity rather than only forecasting accuracy.

Decomposition has also been used for software-style modularity. A trained RNN, including LSTM variants, can be decomposed into per-label binary modules by identifying neurons, gates, and weights associated with each output label, then rewiring the output layer. The reported decomposition cost is small, with Accuracy $X_t$9 and BLEU score $X_r$0, while the resulting modules can be reused and replaced without retraining (Imtiaz et al., 2022). Here DLSTM means decomposition of a trained recurrent system into independently manipulable components.

For formal analysis, functional decomposition rewrites the LSTM cell as a directed acyclic graph of observables, each generated by unary or binary functions, and then constructs hybrid zonotope over-approximations of the input–output graph. In the demonstrated 5-node LSTM example, the final hybrid zonotope for the graph has complexity $X_r$1, $X_r$2, $X_r$3, and $X_r$4 (Glunt et al., 19 Mar 2025). This pushes DLSTM beyond forecasting into reachability-style reasoning about recurrent dynamics under set uncertainty.

6. Limitations, misconceptions, and future directions

The surveyed literature suggests that a common misconception is to treat DLSTM as a single named model. The evidence instead supports a family resemblance: all variants introduce a decomposition that changes what the LSTM has to model, but the decomposition target, mathematical operator, fusion rule, and optimization objective differ substantially across papers. In some cases decomposition is external preprocessing, in others an internal block, a latent-factor model, a parameterization of the weights, a hidden-state factorization, a post-hoc modularization, or an analysis-only transformation.

Several limitations recur. Decomposition choices are sensitive hyperparameters: the Bitcoin VMD-LSTM depends on the number of modes $X_r$5, with the authors examining $X_r$6 and selecting $X_r$7 after residual-energy analysis (Boadi, 11 Sep 2025). Solar flare DLSTM depends on the window size $X_r$8, the 3-hour regularization interval, and resampling ratios such as $X_r$9 and $H_t$0, with the paper explicitly warning that aggressive oversampling can overfit rare events (Hassani et al., 7 Jul 2025). Tensor-factor DLSTMs depend on rank selection: CP performance in spectrum learning improves up to about $H_t$1 and can worsen beyond that, while tensor-train LSTMs require careful choice of core counts and TT ranks, with very high compression degrading perplexity (Joneidi et al., 2019, Murua et al., 2020).

Computational overhead is another recurring issue. VMD-LSTM trains 15 separate LSTMs, increasing cost relative to a single baseline model (Boadi, 11 Sep 2025). Bagged ensemble DLSTM in solar prediction adds multiple bootstrap-trained recurrent models (Hassani et al., 7 Jul 2025). Hybrid-zonotope analysis of LSTMs can generate thousands of generators and binary variables even for a small 5-node network (Glunt et al., 19 Mar 2025). These costs are often traded for denoising, interpretability, or robustness.

Generality is also constrained by domain-specific evaluation. The Bitcoin results are demonstrated on daily BTC-USD closing prices from 2017-08-11 to 2025-06-13 and are not tested on other assets or frequencies (Boadi, 11 Sep 2025). The EV SOC study uses full-lifecycle data from a single vehicle and does not present cross-vehicle validation (Wang et al., 13 May 2026). DeLELSTM explicitly notes that its explanations are limited to individual variables and do not directly model variable interactions (Wang et al., 2023). Spectrum occupancy learning is validated on a large-scale synthetic dataset rather than a real sensing deployment (Joneidi et al., 2019).

The proposed future directions are correspondingly heterogeneous. Suggested extensions include adding attention mechanisms to VMD-LSTM and applying the approach to other financial instruments or medical signals (Boadi, 11 Sep 2025), exploring neural Tucker variants, low-rank tensor recovery, and graph-regularized latent modeling for SOC estimation (Wang et al., 13 May 2026), and integrating predictive architectures into deep reinforcement learning pipelines for trading (Bilokon et al., 2023). Taken together, these proposals reinforce the main historical lesson of DLSTM: decomposition is not an end in itself, but a design strategy for reallocating complexity between representation, recurrence, and recomposition.