# TILDE-Q: A Transformation Invariant Loss Function for Time-Series Forecasting

arXiv: https://arxiv.org/abs/2210.15050 · HTML: https://mlsanity.com/paper/2210.15050/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 59 references that are in the map.
- First posted on arXiv 26 Oct 2022.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting](https://mlsanity.com/paper/2201.12740/) (2022)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [TimesNet: Temporal 2D-Variation Modeling for General Time Series Analysis](https://mlsanity.com/paper/2210.02186/) (2022)
- [N-BEATS: Neural basis expansion analysis for interpretable time series forecasting](https://mlsanity.com/paper/1905.10437/) (2019)
- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [Non-stationary Transformers: Exploring the Stationarity in Time Series Forecasting](https://mlsanity.com/paper/2205.14415/) (2022)
- [Dilated Residual Networks](https://mlsanity.com/paper/1705.09914/) (2017)
- [Soft-DTW: a Differentiable Loss Function for Time-Series](https://mlsanity.com/paper/1703.01541/) (2017)
- [Estimating Counterfactual Treatment Outcomes over Time Through Adversarially Balanced Representations](https://mlsanity.com/paper/2002.04083/) (2020)
- [Mem2Seq: Effectively Incorporating Knowledge Bases into End-to-End Task-Oriented Dialog Systems](https://mlsanity.com/paper/1804.08217/) (2018)
- [Shape and Time Distortion Loss for Training Deep Time Series Forecasting Models](https://mlsanity.com/paper/1909.09020/) (2019)
- [Learning to Remember Rare Events](https://mlsanity.com/paper/1703.03129/) (2017)
- [Differentiable Dynamic Programming for Structured Prediction and Attention](https://mlsanity.com/paper/1802.03676/) (2018)
- [Learning to Remember Patterns: Pattern Matching Memory Networks for Traffic Forecasting](https://mlsanity.com/paper/2110.10380/) (2021)
- [Seq-U-Net: A One-Dimensional Causal U-Net for Efficient Sequence Modelling](https://mlsanity.com/paper/1911.06393/) (2019)
- [Autowarp: Learning a Warping Distance from Unlabeled Time Series Using Sequence Autoencoders](https://mlsanity.com/paper/1810.10107/) (2018)
- [Dynamic State Warping](https://mlsanity.com/paper/1703.01141/) (2017)

## Built on by (most cited first, 8 of 8)

- [DBLoss: Decomposition-based Loss Function for Time Series Forecasting](https://mlsanity.com/paper/2510.23672/) (2025)
- [Patch-wise Structural Loss for Time Series Forecasting](https://mlsanity.com/paper/2503.00877/) (2025)
- [Selective Learning for Deep Time Series Forecasting](https://mlsanity.com/paper/2510.25207/) (2025)
- [KEDformer:Knowledge Extraction Seasonal Trend Decomposition for Long-term Sequence Prediction](https://mlsanity.com/paper/2412.05421/) (2024)
- [LLM-based Evaluation Policy Extraction for Ecological Modeling](https://mlsanity.com/paper/2505.13794/) (2025)
- [CP Loss: Channel-wise Perceptual Loss for Time Series Forecasting](https://mlsanity.com/paper/2601.18829/) (2026)
- [Asymmetric Peak-Aware Loss for Peak-Critical Time Series Forecasting](https://mlsanity.com/paper/2607.14871/) (2026)
- [Not All Variables Agree: Reliability-Aware Variable-Wise Gradient Surgery for Multivariate Time-Series Forecasting](https://mlsanity.com/paper/2609.08554/) (2026)

## Abstract (arXiv)

Time-series forecasting has gained increasing attention in the field of artificial intelligence due to its potential to address real-world problems across various domains, including energy, weather, traffic, and economy. While time-series forecasting is a well-researched field, predicting complex temporal patterns such as sudden changes in sequential data still poses a challenge with current models. This difficulty stems from minimizing Lp norm distances as loss functions, such as mean absolute error (MAE) or mean square error (MSE), which are susceptible to both intricate temporal dynamics modeling and signal shape capturing. Furthermore, these functions often cause models to behave aberrantly and generate uncorrelated results with the original time-series. Consequently, developing a shape-aware loss function that goes beyond mere point-wise comparison is essential. In this paper, we examine the definition of shape and distortions, which are crucial for shape-awareness in time-series forecasting, and provide a design rationale for the shape-aware loss function. Based on our design rationale, we propose a novel, compact loss function called TILDEQ (Transformation Invariant Loss function with Distance EQuilibrium) that considers not only amplitude and phase distortions but also allows models to capture the shape of time-series sequences. Furthermore, TILDE-Q supports the simultaneous modeling of periodic and nonperiodic temporal dynamics. We evaluate the efficacy of TILDE-Q by conducting extensive experiments under both periodic and nonperiodic conditions with various models ranging from naive to state-of-the-art. The experimental results show that the models trained with TILDE-Q surpass those trained with other metrics, such as MSE and DILATE, in various real-world applications, including electricity, traffic, illness, economics, weather, and electricity transformer temperature (ETT).