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Author: MatteoGaetzner

feeds/all-en.atom.xml

@@ -20,7 +20,7 @@ Formally, this can be expressed as</p>
 <p>where the error norm is defined as</p>
 <div class="arithmatex">\[
 |u_\theta - u|_p
-\coloneqq \left( \int_\mathcal{Y} |u(y) - u_\theta(y)|^p , dy \right)^{1/p}.
+\coloneqq \left( \int_\mathcal{Y} |u(y) - u_\theta(y)|^p \; dy \right)^{1/p}.
 \]</div>
 <p>Since the true solution <span class="arithmatex">\(u\)</span> is generally unknown, we cannot compute this loss directly.
 However, we typically know that <span class="arithmatex">\(u\)</span> satisfies a PDE of the form</p>
@@ -38,7 +38,7 @@ This allows us to define the <strong>PDE residual</strong></p>
 <p>To ensure that <span class="arithmatex">\(u_\theta\)</span> satisfies the PDE, we minimize the expected residual:</p>
 <div class="arithmatex">\[
 \mathcal{L}_{\text{PDE}}(\theta)
-\coloneqq \int_\mathcal{Y} |\mathcal{R}_\theta(y)|^p , dy.
+\coloneqq \int_\mathcal{Y} |\mathcal{R}_\theta(y)|^p \; dy.
 \]</div>
 <p>In practice, this integral is approximated by a numerical quadrature over <span class="arithmatex">\(N \in \mathbb{N}\)</span> collocation points <span class="arithmatex">\(\{y_i\}_{i=1}^N\)</span>, yielding</p>
 <div class="arithmatex">\[
@@ -107,7 +107,11 @@ we can include a <strong>data loss</strong></p>
 <div class="arithmatex">\[
 \hat{\theta} = \arg\min_{\theta \in \Theta} \mathcal{L}(\theta),
 \]</div>
-&lt;p&gt;where &lt;span class="arithmatex"&gt;\(\hat{\theta}\)&lt;/span&gt; denotes the parameters obtained by numerical optimization (e.g., using Adam).&lt;/p&gt;</content><category term="Machine Learning"></category><category term="neural-networks"></category><category term="physics"></category><category term="pdes"></category></entry><entry><title>Anytime-Valid Neural Uncertainty Quantification for SPECT Imaging</title><link href="https://www.matteogaetzner.com/anytime-valid-neural-uncertainty-quantification-for-spect-imaging.html" rel="alternate"></link><published>2025-10-11T00:00:00+02:00</published><updated>2025-10-11T00:00:00+02:00</updated><author><name>Matteo Gätzner</name></author><id>tag:www.matteogaetzner.com,2025-10-11:/anytime-valid-neural-uncertainty-quantification-for-spect-imaging.html</id><summary type="html">&lt;p&gt;A short overview of my MSc thesis, which adapts and applies &lt;strong&gt;Sequential and Prior Likelihood Mixing&lt;/strong&gt; (Kirschner et al., 2025) to &lt;strong&gt;anytime-valid uncertainty quantification&lt;/strong&gt; in tomographic imaging. The work explores how to obtain statistically valid uncertainty for &lt;strong&gt;SPECT reconstructions&lt;/strong&gt; using both classical estimators and modern neural predictors such as U-Net ensembles and diffusion models.&lt;/p&gt;</summary><content type="html">&lt;p&gt;&lt;a href="files/thesis.pdf" target="_blank"&gt;Download the thesis (PDF)&lt;/a&gt;&lt;/p&gt;
+&lt;p&gt;where &lt;span class="arithmatex"&gt;\(\hat{\theta}\)&lt;/span&gt; denotes the parameters obtained by numerical optimization (e.g., using Adam).&lt;/p&gt;
+&lt;hr /&gt;
+&lt;h2 id="acknowledgment"&gt;Acknowledgment&lt;/h2&gt;
+&lt;p&gt;This post is based on material covered in the &lt;strong&gt;AI in the Sciences and Engineering (HS 2025)&lt;/strong&gt; lecture by &lt;strong&gt;Prof. Dr. Siddhartha Mishra&lt;/strong&gt;,
+&lt;em&gt;Computational and Applied Mathematics Laboratory (CamLab), Seminar for Applied Mathematics (SAM), D-MATH, ETH AI Center, and Swiss National AI Institute (SNAI), ETH Zürich, Switzerland.&lt;/em&gt;&lt;/p&gt;</content><category term="Machine Learning"></category><category term="neural-networks"></category><category term="physics"></category><category term="pdes"></category></entry><entry><title>Anytime-Valid Neural Uncertainty Quantification for SPECT Imaging</title><link href="https://www.matteogaetzner.com/anytime-valid-neural-uncertainty-quantification-for-spect-imaging.html" rel="alternate"></link><published>2025-10-11T00:00:00+02:00</published><updated>2025-10-11T00:00:00+02:00</updated><author><name>Matteo Gätzner</name></author><id>tag:www.matteogaetzner.com,2025-10-11:/anytime-valid-neural-uncertainty-quantification-for-spect-imaging.html</id><summary type="html">&lt;p&gt;A short overview of my MSc thesis, which adapts and applies &lt;strong&gt;Sequential and Prior Likelihood Mixing&lt;/strong&gt; (Kirschner et al., 2025) to &lt;strong&gt;anytime-valid uncertainty quantification&lt;/strong&gt; in tomographic imaging. The work explores how to obtain statistically valid uncertainty for &lt;strong&gt;SPECT reconstructions&lt;/strong&gt; using both classical estimators and modern neural predictors such as U-Net ensembles and diffusion models.&lt;/p&gt;</summary><content type="html">&lt;p&gt;&lt;a href="files/thesis.pdf" target="_blank"&gt;Download the thesis (PDF)&lt;/a&gt;&lt;/p&gt;
 &lt;hr /&gt;
 &lt;h2 id="abstract"&gt;Abstract&lt;/h2&gt;
 &lt;p&gt;We develop anytime-valid methods for uncertainty quantification in tomographic imaging, with a focus on single-photon emission tomography (SPECT). In SPECT, sequentially acquired data is used to reconstruct images representing the radioactivity distribution inside the object. In addition to producing image reconstructions, our approach constructs &lt;em&gt;confidence sequences&lt;/em&gt;: collections of confidence sets that contain the true but unknown image with high probability simultaneously across all acquisition steps. We investigate two variants: &lt;em&gt;prior likelihood mixing&lt;/em&gt; and &lt;em&gt;sequential likelihood mixing&lt;/em&gt;. Both employ likelihood-based constructions, but differ in how they use user-defined distributions. We parameterize these distributions using classical statistical estimators (MLE, MAP) as well as neural methods, namely U-Net ensembles and diffusion models. In numerical experiments, we simulate SPECT data and compare the tightness and empirical coverage rate of different confidence sequences. Empirically, sequential likelihood mixing proves to be a particularly effective method for constructing confidence sequences. The performance of this method depends on the image predictor used: U-Net ensembles often yield tight and reliable confidence sets, while in some settings classical estimators (MLE, MAP) perform best. We also present strategies for generating uncertainty visualizations. Our results suggest that combining statistical theory with neural predictors enables principled, real-time uncertainty quantification, which may support clinical decision-making in SPECT and related modalities.&lt;/p&gt;

feeds/all.atom.xml

@@ -20,7 +20,7 @@ Formally, this can be expressed as</p>
 <p>where the error norm is defined as</p>
 <div class="arithmatex">\[
 |u_\theta - u|_p
-\coloneqq \left( \int_\mathcal{Y} |u(y) - u_\theta(y)|^p , dy \right)^{1/p}.
+\coloneqq \left( \int_\mathcal{Y} |u(y) - u_\theta(y)|^p \; dy \right)^{1/p}.
 \]</div>
 <p>Since the true solution <span class="arithmatex">\(u\)</span> is generally unknown, we cannot compute this loss directly.
 However, we typically know that <span class="arithmatex">\(u\)</span> satisfies a PDE of the form</p>
@@ -38,7 +38,7 @@ This allows us to define the <strong>PDE residual</strong></p>
 <p>To ensure that <span class="arithmatex">\(u_\theta\)</span> satisfies the PDE, we minimize the expected residual:</p>
 <div class="arithmatex">\[
 \mathcal{L}_{\text{PDE}}(\theta)
-\coloneqq \int_\mathcal{Y} |\mathcal{R}_\theta(y)|^p , dy.
+\coloneqq \int_\mathcal{Y} |\mathcal{R}_\theta(y)|^p \; dy.
 \]</div>
 <p>In practice, this integral is approximated by a numerical quadrature over <span class="arithmatex">\(N \in \mathbb{N}\)</span> collocation points <span class="arithmatex">\(\{y_i\}_{i=1}^N\)</span>, yielding</p>
 <div class="arithmatex">\[
@@ -107,7 +107,11 @@ we can include a <strong>data loss</strong></p>
 <div class="arithmatex">\[
 \hat{\theta} = \arg\min_{\theta \in \Theta} \mathcal{L}(\theta),
 \]</div>
-&lt;p&gt;where &lt;span class="arithmatex"&gt;\(\hat{\theta}\)&lt;/span&gt; denotes the parameters obtained by numerical optimization (e.g., using Adam).&lt;/p&gt;</content><category term="Machine Learning"></category><category term="neural-networks"></category><category term="physics"></category><category term="pdes"></category></entry><entry><title>Anytime-Valid Neural Uncertainty Quantification for SPECT Imaging</title><link href="https://www.matteogaetzner.com/anytime-valid-neural-uncertainty-quantification-for-spect-imaging.html" rel="alternate"></link><published>2025-10-11T00:00:00+02:00</published><updated>2025-10-11T00:00:00+02:00</updated><author><name>Matteo Gätzner</name></author><id>tag:www.matteogaetzner.com,2025-10-11:/anytime-valid-neural-uncertainty-quantification-for-spect-imaging.html</id><summary type="html">&lt;p&gt;A short overview of my MSc thesis, which adapts and applies &lt;strong&gt;Sequential and Prior Likelihood Mixing&lt;/strong&gt; (Kirschner et al., 2025) to &lt;strong&gt;anytime-valid uncertainty quantification&lt;/strong&gt; in tomographic imaging. The work explores how to obtain statistically valid uncertainty for &lt;strong&gt;SPECT reconstructions&lt;/strong&gt; using both classical estimators and modern neural predictors such as U-Net ensembles and diffusion models.&lt;/p&gt;</summary><content type="html">&lt;p&gt;&lt;a href="files/thesis.pdf" target="_blank"&gt;Download the thesis (PDF)&lt;/a&gt;&lt;/p&gt;
+&lt;p&gt;where &lt;span class="arithmatex"&gt;\(\hat{\theta}\)&lt;/span&gt; denotes the parameters obtained by numerical optimization (e.g., using Adam).&lt;/p&gt;
+&lt;hr /&gt;
+&lt;h2 id="acknowledgment"&gt;Acknowledgment&lt;/h2&gt;
+&lt;p&gt;This post is based on material covered in the &lt;strong&gt;AI in the Sciences and Engineering (HS 2025)&lt;/strong&gt; lecture by &lt;strong&gt;Prof. Dr. Siddhartha Mishra&lt;/strong&gt;,
+&lt;em&gt;Computational and Applied Mathematics Laboratory (CamLab), Seminar for Applied Mathematics (SAM), D-MATH, ETH AI Center, and Swiss National AI Institute (SNAI), ETH Zürich, Switzerland.&lt;/em&gt;&lt;/p&gt;</content><category term="Machine Learning"></category><category term="neural-networks"></category><category term="physics"></category><category term="pdes"></category></entry><entry><title>Anytime-Valid Neural Uncertainty Quantification for SPECT Imaging</title><link href="https://www.matteogaetzner.com/anytime-valid-neural-uncertainty-quantification-for-spect-imaging.html" rel="alternate"></link><published>2025-10-11T00:00:00+02:00</published><updated>2025-10-11T00:00:00+02:00</updated><author><name>Matteo Gätzner</name></author><id>tag:www.matteogaetzner.com,2025-10-11:/anytime-valid-neural-uncertainty-quantification-for-spect-imaging.html</id><summary type="html">&lt;p&gt;A short overview of my MSc thesis, which adapts and applies &lt;strong&gt;Sequential and Prior Likelihood Mixing&lt;/strong&gt; (Kirschner et al., 2025) to &lt;strong&gt;anytime-valid uncertainty quantification&lt;/strong&gt; in tomographic imaging. The work explores how to obtain statistically valid uncertainty for &lt;strong&gt;SPECT reconstructions&lt;/strong&gt; using both classical estimators and modern neural predictors such as U-Net ensembles and diffusion models.&lt;/p&gt;</summary><content type="html">&lt;p&gt;&lt;a href="files/thesis.pdf" target="_blank"&gt;Download the thesis (PDF)&lt;/a&gt;&lt;/p&gt;
 &lt;hr /&gt;
 &lt;h2 id="abstract"&gt;Abstract&lt;/h2&gt;
 &lt;p&gt;We develop anytime-valid methods for uncertainty quantification in tomographic imaging, with a focus on single-photon emission tomography (SPECT). In SPECT, sequentially acquired data is used to reconstruct images representing the radioactivity distribution inside the object. In addition to producing image reconstructions, our approach constructs &lt;em&gt;confidence sequences&lt;/em&gt;: collections of confidence sets that contain the true but unknown image with high probability simultaneously across all acquisition steps. We investigate two variants: &lt;em&gt;prior likelihood mixing&lt;/em&gt; and &lt;em&gt;sequential likelihood mixing&lt;/em&gt;. Both employ likelihood-based constructions, but differ in how they use user-defined distributions. We parameterize these distributions using classical statistical estimators (MLE, MAP) as well as neural methods, namely U-Net ensembles and diffusion models. In numerical experiments, we simulate SPECT data and compare the tightness and empirical coverage rate of different confidence sequences. Empirically, sequential likelihood mixing proves to be a particularly effective method for constructing confidence sequences. The performance of this method depends on the image predictor used: U-Net ensembles often yield tight and reliable confidence sets, while in some settings classical estimators (MLE, MAP) perform best. We also present strategies for generating uncertainty visualizations. Our results suggest that combining statistical theory with neural predictors enables principled, real-time uncertainty quantification, which may support clinical decision-making in SPECT and related modalities.&lt;/p&gt;

feeds/machine-learning.atom.xml

@@ -20,7 +20,7 @@ Formally, this can be expressed as</p>
 <p>where the error norm is defined as</p>
 <div class="arithmatex">\[
 |u_\theta - u|_p
-\coloneqq \left( \int_\mathcal{Y} |u(y) - u_\theta(y)|^p , dy \right)^{1/p}.
+\coloneqq \left( \int_\mathcal{Y} |u(y) - u_\theta(y)|^p \; dy \right)^{1/p}.
 \]</div>
 <p>Since the true solution <span class="arithmatex">\(u\)</span> is generally unknown, we cannot compute this loss directly.
 However, we typically know that <span class="arithmatex">\(u\)</span> satisfies a PDE of the form</p>
@@ -38,7 +38,7 @@ This allows us to define the <strong>PDE residual</strong></p>
 <p>To ensure that <span class="arithmatex">\(u_\theta\)</span> satisfies the PDE, we minimize the expected residual:</p>
 <div class="arithmatex">\[
 \mathcal{L}_{\text{PDE}}(\theta)
-\coloneqq \int_\mathcal{Y} |\mathcal{R}_\theta(y)|^p , dy.
+\coloneqq \int_\mathcal{Y} |\mathcal{R}_\theta(y)|^p \; dy.
 \]</div>
 <p>In practice, this integral is approximated by a numerical quadrature over <span class="arithmatex">\(N \in \mathbb{N}\)</span> collocation points <span class="arithmatex">\(\{y_i\}_{i=1}^N\)</span>, yielding</p>
 <div class="arithmatex">\[
@@ -107,4 +107,8 @@ we can include a <strong>data loss</strong></p>
 <div class="arithmatex">\[
 \hat{\theta} = \arg\min_{\theta \in \Theta} \mathcal{L}(\theta),
 \]</div>
-&lt;p&gt;where &lt;span class="arithmatex"&gt;\(\hat{\theta}\)&lt;/span&gt; denotes the parameters obtained by numerical optimization (e.g., using Adam).&lt;/p&gt;</content><category term="Machine Learning"></category><category term="neural-networks"></category><category term="physics"></category><category term="pdes"></category></entry></feed>
+&lt;p&gt;where &lt;span class="arithmatex"&gt;\(\hat{\theta}\)&lt;/span&gt; denotes the parameters obtained by numerical optimization (e.g., using Adam).&lt;/p&gt;
+&lt;hr /&gt;
+&lt;h2 id="acknowledgment"&gt;Acknowledgment&lt;/h2&gt;
+&lt;p&gt;This post is based on material covered in the &lt;strong&gt;AI in the Sciences and Engineering (HS 2025)&lt;/strong&gt; lecture by &lt;strong&gt;Prof. Dr. Siddhartha Mishra&lt;/strong&gt;,
+&lt;em&gt;Computational and Applied Mathematics Laboratory (CamLab), Seminar for Applied Mathematics (SAM), D-MATH, ETH AI Center, and Swiss National AI Institute (SNAI), ETH Zürich, Switzerland.&lt;/em&gt;&lt;/p&gt;</content><category term="Machine Learning"></category><category term="neural-networks"></category><category term="physics"></category><category term="pdes"></category></entry></feed>