1D-to-2D Temporal Transformation

• 1D-to-2D temporal transformation is a process that maps one-dimensional time sequences into two-dimensional arrays to expose hidden patterns and dependencies.
• It leverages techniques like convolutional reshaping, delay embedding, and periodization to enable advanced analysis in neural vocoders, time series forecasting, and symbolic dynamics.
• This method improves modeling efficiency by disentangling local and global temporal variations, allowing more effective use of spatial architectures such as CNNs and Vision Transformers.

A 1D-to-2D temporal transformation is any process by which a one-dimensional temporal sequence or structured signal—typically indexed by time—is systematically reshaped, embedded, or encoded into a two-dimensional array or tensor. This procedure, motivated by representational, computational, or theoretical considerations, enables new kinds of algorithmic, statistical, or dynamical analysis by exposing latent patterns, spatializing temporal dependencies, or leveraging two-dimensional modeling architectures. Such transformations are foundational in modern neural vocoders, time series analysis architectures, dynamical system forecasting, and symbolic dynamics.

1. Formal Definitions and Foundational Contexts

The abstract mechanism of 1D-to-2D temporal transformation varies across domains, but always maintains precise correspondence between temporal structure in the source sequence and the two-dimensional organization in the transformed object.

• In neural vocoding (e.g., iSTFTNet2), a 1D temporal sequence (mel-spectrogram or coarse time-ordered feature) is mapped into a 2D representation with explicit time and frequency axes. This is accomplished via a series of convolutional, upsampling, and reshaping operations, optimized for downstream signal reconstruction fidelity (Kaneko et al., 2023).
• In symbolic dynamics, mappings are constructed such that a 1D subshift (effective closure under shift operator) is embedded as a vertical slice in a 2D subshift of finite type (SFT), with strict constraints ensuring semantic equivalence between the original sequence class and the corresponding column space of the 2D array (Bruno et al., 2010).
• In time series deep learning (e.g., TimesNet, Delayformer), 1D signals are restructured into 2D tensors either by explicit periodization (e.g., stacking cycles of a dominant period) or by delay embedding (e.g., Hankel matrices), making periodicities, dependencies, or nonlinear correlations accessible to 2D filters or attention mechanisms (Wu et al., 2022, Wang et al., 13 Jun 2025).

The goal in each case is to capitalize on the increased expressive power, structural disentanglement, or dynamical regularity afforded by the two-dimensional domain.

2. Mathematical Formulations and Transformation Pipelines

2.1 CNN-Based Audio/Signal Representations

In iSTFTNet2 (Kaneko et al., 2023), the 1D-to-2D transformation proceeds via the following stages:

1. Input Representation:

$\mathbf{M}\in\mathbb{R}^{T\times M}$

where $T$ is the number of frames and $M$ is the frequency axis (e.g., 80).

1. Temporal Upsampling (1D CNN Blocks):
   • Sequential application of ResBlock and transposed conv layers,
   • Expands temporal resolution by $s_t$ (typically 8),
   • Output: $\mathbf{H}^{(L)}\in\mathbb{R}^{C\times T'}$, $T' = 8T$.
2. 1D-to-2D Projection and Reshape:
   • 1×1 convolution projects each time step to a $C'\times F_s$ patch.
   • $$\mathbf{Z}^{(0)}\in\mathbb{R}^{C'\times F_s\times T'}$$ via reshaping.
3. 2D Spectrogram Modeling:
   • 2D ResBlock or ShuffleBlock stacks (typically 3 layers),
   • Kernel: $3\times 3$; processes the $(F_s,t)$ spectrogram.
4. Frequency Upsampling (2D Transposed Conv):
   • Stack of three $(4,1)$ transposed convolutions,
   • Expands frequency to $F = F_s \times 8$.
5. Spectrogram-to-Waveform Synthesis:
   • 1×1 conv yields real-imaginary ($2F$) at each ($f,t$),
   • iSTFT (inverse short-time Fourier transform) recovers $\hat{x}[n]$:

   $$\hat{x}[n] = \sum_{m=0}^{T'-1} \Re\left\{ \sum_{k=0}^{F-1} \hat{\mathbf{S}}[k,m] e^{j\,2\pi k\,n/f_s} \right\} w_s[n-m h_s]$$

2.2 Periodicity-Aware and Delay Embedding Methods

• TimesNet: Reshapes 1D input $x_{1:N}$ into 2D according to discovered periods (Wu et al., 2022):

$$X^{(i)}_{2D} = \mathrm{Reshape}_{(M_i, L_i)}(x_{1:\widetilde N_i})$$

Columns correspond to intra-period points; rows to corresponding phase across cycles.

• Delayformer mvSTI: Forms a Hankel (delay-embedded) matrix for each variable (Wang et al., 13 Jun 2025):

$Y_k^{(t)}\in\mathbb{R}^{L\times m}$

where rows are lagged values, columns index the forecast horizon; 2D "images" are subsequently processed via patching and a shared Vision Transformer encoder.

2.3 Symbolic Dynamics and Tiling

• Vertical embedding: The vertical slice $X(i,j) = X(i,j+1)$, row $X(\cdot,0)\in S$, carries the 1D effective subshift $S$ into the 2D SFT context.

• Stripe and fixed-point constructions (Bruno et al., 2010):

  • Aubrun–Sablik: Uses a geometric decomposition of the plane into vertical stripes of dyadically increasing width, each simulating a computation checking forbidden subwords.
  • Durand–Romashchenko–Shen: Self-similar macro-tile hierarchy, each macrotile simulating the same computation recursively at all scales.

3. Architectural and Representation-Theoretic Motivations

1D-to-2D transformations are adopted when the original 1D representation is suboptimal for statistical modeling or theoretical reasoning. Key motivations include:

• Enhanced Expressivity: 2D convolution or self-attention jointly exploits dependencies along both axes—temporal locality and pseudo-spatial grouping—enabling richer pattern mining than possible in 1D (Kaneko et al., 2023, Wu et al., 2022).
• Disentanglement of Variation: Decomposition into intra- and inter-period variation clarifies representation of multi-scale and multivariate dependencies in time series data. Multi-period reshaping isolates local (e.g., within period) versus global (e.g., across cycles) factors (Wu et al., 2022).
• Leveraging Spatial Architectures: The emergence of highly parameter-efficient and scalable 2D inductive biases (e.g., CNNs, Vision Transformers) motivates mapping temporal signals to a spatial structure for more effective learning (Wang et al., 13 Jun 2025).
• Facilitating Theoretical Simulation: In symbolic dynamics, mapping complex 1D subshifts to 2D allows their simulation within the framework of SFTs, where local rules, computational hierarchy, and error control are more tractable (Bruno et al., 2010).

4. Applications in Signal Processing, Time Series, and Dynamics

Neural Vocoding and Audio Synthesis

The iSTFTNet2 pipeline demonstrates that 1D-to-2D transformation enables fast, lightweight, and high-fidelity waveform synthesis by:

• Upsampling time via 1D CNN blocks,
• Mapping to a compact 2D time-frequency representation,
• Employing 2D blocks for local spectro-temporal modeling,
• Restoring full resolution before iSTFT waveform reconstruction,
• Training with composite adversarial (LSGAN), feature-matching, and mel-spectrogram objective terms to ensure both temporal and spectral fidelity (Kaneko et al., 2023).

Time Series Analysis

TimesNet introduces a period-based 1D-to-2D mapping:

• Data-driven adaptive discovery of multiple periodicities via FFT amplitude peaks,
• Reshaping into 2D tensors indexed by period and cycle count,
• 2D Inception-style convolution to extract complex interactions between varying temporal scales,
• Demonstrated SOTA performance across forecasting, imputation, classification, and anomaly detection tasks, with ablation studies confirming significant improvements from the 2D step (Wu et al., 2022).

Delayformer applies delay-embedding theory:

• Forms Hankel matrices for each channel,
• Patches and encodes these via a single shared ViT to leverage both local and global relationships within the 2D "state image",
• Decodes per variable, recovering multi-step forecasts with high fidelity,
• Empirically robust against noise and high-dimensionality, outperforming SOTA methods in several domains (Wang et al., 13 Jun 2025).

Symbolic Dynamics

Two constructive approaches embed any effectively closed 1D subshift into a 2D SFT:

• The Aubrun–Sablik (stripe) and Durand–Romashchenko–Shen (fixed-point) methods rigorously define how specific vertical slices of the 2D tiling exactly recover the 1D subshift, ensuring dynamical invariants and complexity properties are preserved (Bruno et al., 2010).

5. Algorithmic and Computational Properties

The design specifics of 1D-to-2D pipelines influence computational resource requirements and algorithmic trade-offs.

Method	Reshaping Mechanism	2D Modeling Module
iSTFTNet2 (Kaneko et al., 2023)	1D CNN time-up, 1x1 proj to (freq,time)	2D CNN blocks, transposed conv
TimesNet (Wu et al., 2022)	FFT-driven periodization and reshape	2D Inception-Block CNN
Delayformer (Wang et al., 13 Jun 2025)	Hankelization (delay-embed) per channel	Shared ViT on patches
Subshift Tiling (Bruno et al., 2010)	Symbolic vertical embedding	Macro-tile or stripe SFT rules

Considerations include:

• Dimensionality and computational cost: 2D representations incur higher computation, but allow kernel sharing and parallelism.
• Parameter sharing: Convolutional and attention-based blocks can exploit tensor structure efficiently.
• Gradient propagation: Differentiable reshaping and projection enable seamless integration into backprop pipelines (as in neural architectures).
• Data alignment: Zero-padding, frequency rounding, and truncation influence how temporal boundaries map into the 2D structure.
• Complexity bounds: In symbolic coding, the tile set cardinality and neighborhood radius scale polynomially with the complexity of the original 1D constraint (machine size and forbidden list length).

6. Theoretical Implications and Open Problems

• Universality: It is formally proven that any effectively closed 1D subshift can be simulated as the vertical projection of a 2D SFT, with preservation of entropy and other dynamical invariants (Bruno et al., 2010).
• Expressive power: Empirical ablations demonstrate improved learning performance via the 2Dization in neural models; removing 2D blocks degrades results significantly (Wu et al., 2022).
• Optimization: The design space for dividing periodicity, choosing patch or frequency axis size, and balancing network complexity remains open, with practical trade-offs observed in empirical studies.
• Error correction and redundancy: Symbolic constructive methods incorporating error-correction schemes (not present in all frameworks) offer potential robustness for 1D-to-2D temporal embeddings in symbolic and potentially neural systems.
• Open complexity-theoretic questions: Open problems include minimizing tile set overhead, exploring further dimension reduction, and the computability of factorizations between high-dimensional SFTs and effective 1D subshifts (Bruno et al., 2010).

7. Empirical and Comparative Performance

Across empirical studies in forecasting, imputation, and anomaly classification:

• TimesNet demonstrates consistent SOTA metrics across a large suite of benchmarks, with the 1D-to-2D temporal transformation shown to provide categorical improvements over strict 1D representations, as quantified in ablation experiments (Wu et al., 2022).
• Delayformer surpasses existing approaches in high-dimensional prediction tasks via spatiotemporal fusion explicitly enabled by the delay-embedded 2D transformation and ViT encoding (Wang et al., 13 Jun 2025).
• iSTFTNet2 achieves lighter and faster temporal-to-frequency modeling without quality loss compared to 1D-only CNNs, preserving both holistic and local spectral fidelity (Kaneko et al., 2023).

A plausible implication is that the systematic design and adaptation of 1D-to-2D temporal transformations, customized for application context, will remain a central tool in both practice and theory for a wide spectrum of time-structured modeling challenges.