Estimating Probability Densities with Transformer and Denoising Diffusion (2407.15703v1)

Abstract: Transformers are often the go-to architecture to build foundation models that ingest a large amount of training data. But these models do not estimate the probability density distribution when trained on regression problems, yet obtaining full probabilistic outputs is crucial to many fields of science, where the probability distribution of the answer can be non-Gaussian and multimodal. In this work, we demonstrate that training a probabilistic model using a denoising diffusion head on top of the Transformer provides reasonable probability density estimation even for high-dimensional inputs. The combined Transformer+Denoising Diffusion model allows conditioning the output probability density on arbitrary combinations of inputs and it is thus a highly flexible density function emulator of all possible input/output combinations. We illustrate our Transformer+Denoising Diffusion model by training it on a large dataset of astronomical observations and measured labels of stars within our Galaxy and we apply it to a variety of inference tasks to show that the model can infer labels accurately with reasonable distributions.

• The paper introduces a novel integrated Transformer and Denoising Diffusion model that captures complex probability densities in high-dimensional tabular data.
• It employs an encoder-only Transformer with a DDPM head to effectively model non-Gaussian, multimodal distributions, validated on astronomical datasets.
• The research demonstrates accurate conditional density and uncertainty estimation, enabling scalable probabilistic regression for scientific applications.

Estimating Probability Densities of Tabular Data using a Transformer Model combined with Denoising Diffusion

The research paper titled "Estimating Probability Densities of Tabular Data using a Transformer Model combined with Denoising Diffusion" introduces a novel approach for modeling probability densities in high-dimensional data using an integrated Transformer and Denoising Diffusion Probabilistic Model (DDPM). This methodology addresses a critical challenge in regression tasks where traditional Transformer models fall short by only predicting scalar values without capturing the probability distribution of outputs.

Introduction

The Transformer architecture has established itself as a cornerstone for constructing foundation models, particularly in natural language processing. However, its application in regression tasks is often limited by its inability to directly estimate the underlying probability density, which is essential for fields like astronomy where the data distribution can be non-Gaussian and multimodal. This paper proposes the integration of a DDPM on top of a Transformer, aiming to provide comprehensive probabilistic outputs for high-dimensional inputs, thereby enhancing the model's flexibility and practical utility in scientific datasets.

Model Architecture

The proposed model consists of an encoder-only Transformer, akin to BERT, which reduces computational overhead by eliminating the decoder component. This aspect is pivotal for scalability, allowing the utilization of extended context sizes during inference without the constraints imposed by positional encoding. The model architecture is showcased in Figure 1 of the paper, which illustrates the flow from the Transformer encoder to the DDPM head, responsible for transforming the hidden states into a probability density distribution.

Denoising Diffusion Probabilistic Models (DDPM)

DDPMs are chosen over Normalizing Flows (NF) due to their inherent ability to model diverse distributions with a singular framework. The DDPM adds noise to samples from the training dataset incrementally, and the model's task is to reverse this process, thereby gradually converting noisy samples back to their denoised states. This mechanism allows the generation of new samples by denoising from a Gaussian distribution, controlled by conditions derived from the Transformer's hidden states.

Training and Implementation

The model is trained on a dataset comprising Gaia DR3 photometry and XP spectra, supplemented by 2MASS photometry and stellar parameters from APOGEE DR17. The training process involves randomly selecting subsets of data as input and outputs in each batch, ensuring the model learns to infer from arbitrary combinations of features. Hyperparameters include a total of 3.7 million parameters, of which 1.8 million are allocated to the DDPM head, and the model is optimized using AdamW with a CosineAnnealingWarmRestarts scheduler over 10240 epochs.

Results

The results demonstrate the model's capability to infer probability densities accurately. Key tests include:

1. Distribution Recovery without Conditions: The model successfully replicates the training set distributions for various stellar parameters as shown in Figure 2.
2. Conditional Distributions: The model accurately captures the conditional distributions of stellar parameters given subsets of input data, as illustrated in Figure 3.
3. Prediction Accuracy and Uncertainty: The inference of median and standard deviation aligns well with ground truth values, validating the model's uncertainty estimation, exemplified in Figure 4.
4. Quantile Distribution: The predicted distributions show uniform quantile placement of ground truth values, indicating robust uncertainty quantification, as evidenced in Figure 5.

Implications and Future Work

This research bridges a significant gap in regression tasks by incorporating probabilistic modeling within a Transformer framework, enhancing the model's interpretability and applicative value in scientific domains. Practical applications extend beyond astronomy to any field requiring non-Gaussian and multimodal density estimation.

Future advancements may include extending the model to higher-dimensional probability distributions by leveraging the sequential generation capability of the Transformer. Additionally, applying this methodology to larger, more diverse datasets would further validate and improve the model's robustness and scalability.

Conclusion

The integration of a DDPM with a Transformer encoder for estimating probability densities represents a substantial methodological improvement for handling high-dimensional regression tasks. Through rigorous validation on astronomical data, this paper showcases how such a combined model can offer flexible, accurate, and comprehensive probabilistic outputs, thus broadening the applicability of foundation models in scientific and high-dimensional data contexts.