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A deep neural network framework for dynamic multi-valued mapping estimation and its applications

Geng Li, Di Qiu, Lok Ming Lui

TL;DR

The paper tackles uncertainty-driven modeling by introducing dynamic multi-valued mappings (DMM), where each input maps to a finite set of plausible outputs with associated probabilities. It proposes a deep neural network framework that jointly learns a generative mapping and a probability predictor using a discrete codebook, with a cluster mapper guiding output selection. A specialized loss combining reconstruction, covariance-based code separation, and an ETF-based classifier enables diverse, non-redundant outputs and calibrated uncertainty, even on imbalanced data. The approach is validated on synthetic shape reconstruction and real imaging problems (lung opacity segmentation and LIDC-IDRI CT scans), showing accurate multi-modal outputs with meaningful uncertainty estimates and competitive GED metrics. This framework offers a practical pathway for reliable multi-solution inference in medical imaging and related domains.

Abstract

This paper addresses the problem of modeling and estimating dynamic multi-valued mappings. While most mathematical models provide a unique solution for a given input, real-world applications often lack deterministic solutions. In such scenarios, estimating dynamic multi-valued mappings is necessary to suggest different reasonable solutions for each input. This paper introduces a deep neural network framework incorporating a generative network and a classification component. The objective is to model the dynamic multi-valued mapping between the input and output by providing a reliable uncertainty measurement. Generating multiple solutions for a given input involves utilizing a discrete codebook comprising finite variables. These variables are fed into a generative network along with the input, producing various output possibilities. The discreteness of the codebook enables efficient estimation of the output's conditional probability distribution for any given input using a classifier. By jointly optimizing the discrete codebook and its uncertainty estimation during training using a specially designed loss function, a highly accurate approximation is achieved. The effectiveness of our proposed framework is demonstrated through its application to various imaging problems, using both synthetic and real imaging data. Experimental results show that our framework accurately estimates the dynamic multi-valued mapping with uncertainty estimation.

A deep neural network framework for dynamic multi-valued mapping estimation and its applications

TL;DR

The paper tackles uncertainty-driven modeling by introducing dynamic multi-valued mappings (DMM), where each input maps to a finite set of plausible outputs with associated probabilities. It proposes a deep neural network framework that jointly learns a generative mapping and a probability predictor using a discrete codebook, with a cluster mapper guiding output selection. A specialized loss combining reconstruction, covariance-based code separation, and an ETF-based classifier enables diverse, non-redundant outputs and calibrated uncertainty, even on imbalanced data. The approach is validated on synthetic shape reconstruction and real imaging problems (lung opacity segmentation and LIDC-IDRI CT scans), showing accurate multi-modal outputs with meaningful uncertainty estimates and competitive GED metrics. This framework offers a practical pathway for reliable multi-solution inference in medical imaging and related domains.

Abstract

This paper addresses the problem of modeling and estimating dynamic multi-valued mappings. While most mathematical models provide a unique solution for a given input, real-world applications often lack deterministic solutions. In such scenarios, estimating dynamic multi-valued mappings is necessary to suggest different reasonable solutions for each input. This paper introduces a deep neural network framework incorporating a generative network and a classification component. The objective is to model the dynamic multi-valued mapping between the input and output by providing a reliable uncertainty measurement. Generating multiple solutions for a given input involves utilizing a discrete codebook comprising finite variables. These variables are fed into a generative network along with the input, producing various output possibilities. The discreteness of the codebook enables efficient estimation of the output's conditional probability distribution for any given input using a classifier. By jointly optimizing the discrete codebook and its uncertainty estimation during training using a specially designed loss function, a highly accurate approximation is achieved. The effectiveness of our proposed framework is demonstrated through its application to various imaging problems, using both synthetic and real imaging data. Experimental results show that our framework accurately estimates the dynamic multi-valued mapping with uncertainty estimation.
Paper Structure (17 sections, 2 theorems, 24 equations, 17 figures, 2 tables, 2 algorithms)

This paper contains 17 sections, 2 theorems, 24 equations, 17 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contribution
  3. Proposed model
  4. Mathematical formulation of dynamic multi-valued mapping problem
  5. Deep neural network framework for DMM problem
  6. Overall Network structure
  7. Optimization of discrete codebook $C$
  8. Probability prediction
  9. Loss Function
  10. Numerical algorithm
  11. Experiments
  12. Experimental setup
  13. Synthetic examples: Shape reconstruction
  14. Lung segmentation with pulmonary opacity
  15. Real applications
  16. ...and 2 more sections

Key Result

theorem 1

Let $v_1, v_2, ..., v_m$ be unit vectors in $\mathbb{R}^n$ such that $\|<v_i,v_j>\| \leq An^{-1/2}$ for all distinct $i,j$, $\frac{1}{2} \leq A \leq \frac{\sqrt{n}}{2}$, then we have $m\leq(\frac{Cn}{A^2})^{CA^2}$ for some absolute constant $C$.

Figures (17)

  • Figure 1: Some samples of lung CT scans. The first column is lung CT scans and the left column is labels from four experts. Experts provide different annotations for each CT scan, resulting in varying numbers and probabilities of outputs.
  • Figure 2: The architecture of our model in the training process.
  • Figure 3: The architecture of our framework for modeling DMM.
  • Figure 4: Results visualization for shape reconstruction. The first row shows the input samples and their labels, and the next row shows the predictions from our method. The probability for each prediction is annotated in the upper-left corner.
  • Figure 5: Results from our model on the shape reconstruction task. Results with the predicted uncertainties ($>1e^{-5}$) are shown.
  • ...and 12 more figures

Theorems & Definitions (5)

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  • theorem 1
  • theorem 2
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