11 research outputs found
Your diffusion model secretly knows the dimension of the data manifold
In this work, we propose a novel framework for estimating the dimension of
the data manifold using a trained diffusion model. A trained diffusion model
approximates the gradient of the log density of a noise-corrupted version of
the target distribution for varying levels of corruption. If the data
concentrates around a manifold embedded in the high-dimensional ambient space,
then as the level of corruption decreases, the score function points towards
the manifold, as this direction becomes the direction of maximum likelihood
increase. Therefore, for small levels of corruption, the diffusion model
provides us with access to an approximation of the normal bundle of the data
manifold. This allows us to estimate the dimension of the tangent space, thus,
the intrinsic dimension of the data manifold. Our method outperforms linear
methods for dimensionality detection such as PPCA in controlled experiments.Comment: arXiv admin note: text overlap with arXiv:2207.0978
Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equation
Score-based diffusion models have emerged as one of the most promising
frameworks for deep generative modelling, due to their state-of-the art
performance in many generation tasks while relying on mathematical foundations
such as stochastic differential equations (SDEs) and ordinary differential
equations (ODEs). Empirically, it has been reported that ODE based samples are
inferior to SDE based samples. In this paper we rigorously describe the range
of dynamics and approximations that arise when training score-based diffusion
models, including the true SDE dynamics, the neural approximations, the various
approximate particle dynamics that result, as well as their associated
Fokker--Planck equations and the neural network approximations of these
Fokker--Planck equations. We systematically analyse the difference between the
ODE and SDE dynamics of score-based diffusion models, and link it to an
associated Fokker--Planck equation. We derive a theoretical upper bound on the
Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms
of a Fokker--Planck residual. We also show numerically that conventional
score-based diffusion models can exhibit significant differences between ODE-
and SDE-induced distributions which we demonstrate using explicit comparisons.
Moreover, we show numerically that reducing the Fokker--Planck residual by
adding it as an additional regularisation term leads to closing the gap between
ODE- and SDE-induced distributions. Our experiments suggest that this
regularisation can improve the distribution generated by the ODE, however that
this can come at the cost of degraded SDE sample quality
Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equation
Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality
Common pitfalls and recommendations for using machine learning to detect and prognosticate for COVID-19 using chest radiographs and CT scans
Abstract: Machine learning methods offer great promise for fast and accurate detection and prognostication of coronavirus disease 2019 (COVID-19) from standard-of-care chest radiographs (CXR) and chest computed tomography (CT) images. Many articles have been published in 2020 describing new machine learning-based models for both of these tasks, but it is unclear which are of potential clinical utility. In this systematic review, we consider all published papers and preprints, for the period from 1 January 2020 to 3 October 2020, which describe new machine learning models for the diagnosis or prognosis of COVID-19 from CXR or CT images. All manuscripts uploaded to bioRxiv, medRxiv and arXiv along with all entries in EMBASE and MEDLINE in this timeframe are considered. Our search identified 2,212 studies, of which 415 were included after initial screening and, after quality screening, 62 studies were included in this systematic review. Our review finds that none of the models identified are of potential clinical use due to methodological flaws and/or underlying biases. This is a major weakness, given the urgency with which validated COVID-19 models are needed. To address this, we give many recommendations which, if followed, will solve these issues and lead to higher-quality model development and well-documented manuscripts
Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equation
Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality
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The impact of imputation quality on machine learning classifiers for datasets with missing values
Background: Classifying samples in incomplete datasets is a common aim for machine learning practitioners, but is non-trivial. Missing data is found in most real-world datasets and these missing values are typically imputed using established methods, followed by classification of the now complete samples. The focus of the machine learning researcher is to optimise the classifier’s performance. Methods: We utilise three simulated and three real-world clinical datasets with different feature types and missingness patterns. Initially, we evaluate how the downstream classifier performance depends on the choice of classifier and imputation methods. We employ ANOVA to quantitatively evaluate how the choice of missingness rate, imputation method, and classifier method influences the performance. Additionally, we compare commonly used methods for assessing imputation quality and introduce a class of discrepancy scores based on the sliced Wasserstein distance. We also assess the stability of the imputations and the interpretability of model built on the imputed data. Results: The performance of the classifier is most affected by the percentage of missingness in the test data, with a considerable performance decline observed as the test missingness rate increases. We also show that the commonly used measures for assessing imputation quality tend to lead to imputed data which poorly matches the underlying data distribution, whereas our new class of discrepancy scores performs much better on this measure. Furthermore, we show that the interpretability of classifier models trained using poorly imputed data is compromised. Conclusions: It is imperative to consider the quality of the imputation when performing downstream classification as the effects on the classifier can be considerable
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The impact of imputation quality on machine learning classifiers for datasets with missing values.
BACKGROUND: Classifying samples in incomplete datasets is a common aim for machine learning practitioners, but is non-trivial. Missing data is found in most real-world datasets and these missing values are typically imputed using established methods, followed by classification of the now complete samples. The focus of the machine learning researcher is to optimise the classifier's performance. METHODS: We utilise three simulated and three real-world clinical datasets with different feature types and missingness patterns. Initially, we evaluate how the downstream classifier performance depends on the choice of classifier and imputation methods. We employ ANOVA to quantitatively evaluate how the choice of missingness rate, imputation method, and classifier method influences the performance. Additionally, we compare commonly used methods for assessing imputation quality and introduce a class of discrepancy scores based on the sliced Wasserstein distance. We also assess the stability of the imputations and the interpretability of model built on the imputed data. RESULTS: The performance of the classifier is most affected by the percentage of missingness in the test data, with a considerable performance decline observed as the test missingness rate increases. We also show that the commonly used measures for assessing imputation quality tend to lead to imputed data which poorly matches the underlying data distribution, whereas our new class of discrepancy scores performs much better on this measure. Furthermore, we show that the interpretability of classifier models trained using poorly imputed data is compromised. CONCLUSIONS: It is imperative to consider the quality of the imputation when performing downstream classification as the effects on the classifier can be considerable
Recommended from our members
The impact of imputation quality on machine learning classifiers for datasets with missing values.
BACKGROUND: Classifying samples in incomplete datasets is a common aim for machine learning practitioners, but is non-trivial. Missing data is found in most real-world datasets and these missing values are typically imputed using established methods, followed by classification of the now complete samples. The focus of the machine learning researcher is to optimise the classifier's performance. METHODS: We utilise three simulated and three real-world clinical datasets with different feature types and missingness patterns. Initially, we evaluate how the downstream classifier performance depends on the choice of classifier and imputation methods. We employ ANOVA to quantitatively evaluate how the choice of missingness rate, imputation method, and classifier method influences the performance. Additionally, we compare commonly used methods for assessing imputation quality and introduce a class of discrepancy scores based on the sliced Wasserstein distance. We also assess the stability of the imputations and the interpretability of model built on the imputed data. RESULTS: The performance of the classifier is most affected by the percentage of missingness in the test data, with a considerable performance decline observed as the test missingness rate increases. We also show that the commonly used measures for assessing imputation quality tend to lead to imputed data which poorly matches the underlying data distribution, whereas our new class of discrepancy scores performs much better on this measure. Furthermore, we show that the interpretability of classifier models trained using poorly imputed data is compromised. CONCLUSIONS: It is imperative to consider the quality of the imputation when performing downstream classification as the effects on the classifier can be considerable
The impact of imputation quality on machine learning classifiers for datasets with missing values
Classifying samples in incomplete datasets is a common aim for machine learning practitioners, but is non-trivial. Missing data is found in most real-world datasets and these missing values are typically imputed using established methods, followed by classification of the now complete samples. The focus of the machine learning researcher is to optimise the classifier’s performance.Peer reviewe