Jakob Zimmermann

Research

Current

Free University Berlin – Discrete Geometry Group

With professor Georg Loho, I work on the mathematical foundations of Deep Learning. We investigate the geometry of the Newton polytopes of neural networks and develop new Explainable AI (XAI) methods from the Difference-of-Convex decomposition of networks. Using agentic proof search in Lean 4, we study loss landscapes of shallow ReLU networks — where skip connections remove spurious minima and how teacher parameters determine the critical points.

Current

Fraunhofer HHI – Berlin

With professor Wojciech Samek I am working on new backpropagation based XAI methods for the Transformer Architecture.

Previously

Fraunhofer IOSB – Karlsruhe

I have worked on runtime monitoring of image recognition models and representation learning. I developed a monitoring approach that is knowledge-guided and works on internal representations of image recognition models and embedding models such as DINO.

Goals

I want to understand neural networks at the level where their computations become mathematically inspectable: as piecewise-linear and geometric objects. My goal is to turn this structure into methods that make modern models more transparent, more monitorable, and more reliable in deployment.

Concretely, I work on three connected questions:

What structure is hidden inside a trained network? I study decompositions of networks into monotone and convex parts, and reformulations of the backward pass as two-player games on an extended network graph — both expose how positive and negative evidence propagate through a model.
When can explanations be trusted? I develop attribution methods whose behaviour is tied to the network’s actual computation rather than only to visual plausibility.
How can models recognise their own failures? I investigate monitoring methods that detect safety-critical perception failures by comparing internal model representations with foundation-model embeddings.

This puts my work at the interface of discrete geometry, explainable AI, and safety: I use Newton polytopes, difference-of-convex decompositions, and two-player games on computational graphs to derive explanations, certificates, and failure signals from the mathematical structure neural networks already have.

Publications

Hidden Monotonicity: Explaining Deep Neural Networks via their DC Decomposition

Jakob Paul Zimmermann and Georg Loho

CVPR 2026 Highlight

Poster Video arXiv PDF

DC decomposition of a ReLU network into positive and negative monotone components

We propose an algorithm to decompose any ReLU Neural Network (CNN, MLP, ResNet) into a difference of two monotone and convex Maxout networks and to stabilize the forward and backward pass through this decomposition. This provides a principled approach for Explainable AI, enabling separate analysis of positive and negative contributions to the network output. Our proposed saliency methods – SplitCAM and SplitLRP – improve on SOTA results on both VGG16 and ResNet18 on ImageNet-S across all Quantus saliency metric categories.

Knowledge-Guided Failure Prediction: Detecting When Object Detectors Miss Safety-Critical Objects

Jakob Paul Zimmermann, Gerrit Holzbach, and David Lerch

SAIAD Workshop at CVPR 2026

arXiv PDF

Dual-encoder architecture of KGFP: YOLO and DINO features are fused through cross-attention and projected into a shared embedding space

We introduce KGFP, a representation-based monitoring framework for detecting when an object detector is likely to miss safety-critical objects. KGFP measures semantic misalignment between YOLOv8 internal features and DINO foundation-model embeddings using a dual-encoder architecture with an angular distance metric. On COCO person detection, KGFP improves person recall from 64.3% to 84.5% at 5% False Positive Rate.

Induced Turán problem in bipartite graphs

Maria Axenovich and Jakob Zimmermann

Discrete Applied Mathematics, Volume 360, Pages 497–505, January 2025

arXiv PDF

Preprints

Removing spurious minima for planar features by skip connections

Jakob Paul Zimmermann, Moritz Grillo, Andrei Balakin, and Georg Loho

arXiv:2610.01728, October 2026

Preprint

arXiv PDF Official repo

Understanding loss landscapes is central to explaining neural-network training, yet their structure remains only partially understood even in simple models. We study the Gaussian population loss of shallow, bias-free ReLU networks in the teacher–student setting. This provides a simple model for studying essential aspects such as feature learning and overparameterization. For teacher networks with positive output weights and planar features, we show that including a learned linear skip removes all spurious local minima with non-negative student output weights once the student network is at least as wide as the teacher network. In contrast, without the skip, we construct a fixed teacher network with positive output weights and only three hidden neurons in input dimension two whose spurious local minima persist at every student width at least three. Thus, a learned linear skip can remove spurious minima that persist under arbitrary overparameterization.

\(P\)\(P_c\)

Network Newton polytopes. A ReLU network \(x\mapsto\sum_i s_i\,\mathrm{ReLU}(\langle w_i,x\rangle)\) with weights \(s_i\ge 0\) is the support function of the zonotope \(P=\sum_i s_i[0,w_i]\), its Newton polytope: the scaled first-layer weights \(s_iw_i\) are its generators. The learned linear skip only translates it: drag \(P\); it snaps to the centered \(P_c=\sum_i s_i[-\tfrac12 w_i,\tfrac12 w_i]\) at the origin and back to its Steiner point \(\tau=\tfrac12\sum_i s_iw_i\).

\(f_P\)\(f_{P_c}\)\(\langle\tau,u\rangle\)

Support functions. The support function of a convex body \(K\) is \(f_K(u)=\max_{v\in K}\langle v,u\rangle\). Translating by \(\tau\) adds exactly the odd term: \(f_P=f_{P_c}+\langle\tau,u\rangle\). Show the vertex cosines \(\langle v,u\rangle\) whose maximum it is, and the switches of the maximising vertex.

Furthermore, we show that a positive output weight student network always learns the subspace spanned by the teacher features: student features at local minima with non-negative student output weights lie in the span of the teacher features. For ReLU networks in two dimensions, even heavily overparameterized student networks have effective width controlled by the teacher width: every critical point with positive student output weights has at most twice as many distinct student feature directions as teacher neurons. Finally, we transfer the benignity result to empirical minima over parameter balls of any prescribed radius, with the required sampling accuracy depending on that radius. Every theorem is formalized in Lean 4.

Playing the Network Backward: A Game Theoretic Attribution Framework

Jakob Paul Zimmermann, Jim Berend, Georg Loho, Sebastian Lapuschkin, and Wojciech Samek

arXiv:2605.06212, May 2026

Preprint

arXiv PDF

Two-player backward game on the extended computational graph: at each state the players decide stop-or-go and where to route, and the induced trajectory distribution recovers the attribution

Backward attribution methods — gradients, LRP, and transformer-specific rules — are usually presented as a zoo of recipes with method-specific stabilisers and no common comparison framework. Building on the ReLU Net Game of Gaubert and Vlassopoulos, we lift the backward pass through a network into a two-player game on an extended computational graph. The game relates to a novel Stopping Decomposition that splits weights by sign into W = W⁺ − W⁻ — a sign-based weight decomposition related to the DC decomposition used directly for attribution in our Hidden Monotonicity work — and unravels excitatory and inhibitory contributions throughout the network. At each state the two players answer (1) stop or go? and (2) where to route?; their optimal policies induce a trajectory distribution, and a weighted trajectory integral recovers the attribution. Game-theoretic ideas — risk aversion, action-space augmentation, policy regularisation — map back to principled adaptations of classical backpropagation rules and boost their performance: one such adaptation outperforms all prior transformer-specific backward methods on ViT-B/16 across localisation metrics.

Bipartite Turán problem on cographs

Jakob Zimmermann

arXiv:2601.07406, January 2026

Preprint

arXiv PDF

K_2,3-free extremal cograph on 18 vertices — drag to explore

We investigate the bipartite Turán problem on cographs and fully classify the edge-maximal cographs with respect to not containing certain bicliques. These extremal cographs turn out to be star-shaped, highly symmetrical and beautiful. Explore more extremal cographs →

Projects

The two-teacher census: every critical point of a planar two-student ReLU landscape

Explore Code

The eight critical points of the Gaussian population loss for one two-neuron teacher ReLU network and a two-neuron student: the exact fit, two collided and two separated spurious local minima, and three saddles, each drawn as teacher and student generators

The smallest non-trivial teacher–student setting one can write down — networks with two hidden ReLU neurons and planar Gaussian input — already has a surprisingly rich loss landscape: the example teacher network gives the student network eight critical points, four of them spurious local minima. We found closed formulas for every critical point of every such teacher, for the plain ReLU network and for the network with a learned linear skip, and mapped the teacher parameters by the local minima they produce (the census). This project yielded a good test bed for the theorems on wider networks and higher input dimensions in the paper. The interactive page lets you drag the teacher and watch all critical points respond.

Extremal cograph viewer

Interactive browser for the edge-maximal K_s,t-free cographs of the cograph preprint

Explore Code

Talks & Conferences

Past

22 Jan 2026

Talk “Bipartite Turán problem on cographs” at Szabó’s Research Seminar, Free University Berlin

17 Mar 2026

Poster “Hidden Monotonicity” at Workshop on Polyhedral Geometry for Neural Networks, Nürnberg

04 Jun 2026

Talk “Knowledge-Guided Failure Prediction” at SAIAD Workshop at CVPR 2026, Denver

07 Jun 2026

Poster “Hidden Monotonicity” at CVPR 2026, Denver — Highlight Poster Video

19 Jun 2026

Posters “Hidden Monotonicity” and “Knowledge-Guided Failure Prediction” at the dida conference, Berlin

01 Jul 2026

Talk “From DC Decompositions to Two-Player Games: A Structural Approach to Explainability in Deep Learning” at Gottschalk's Research Seminar, TU Berlin (IMOS)

15–18 Sep 2026

Poster “From DC Decompositions to Two-Player Games: A Structural Approach to Explainability in Deep Learning” at the ELLIS Summer School on Machine Learning & Computer Vision, ELLIS Munich Unit, TUM Garching

Mathematical Interests

Stochastics Statistical Learning Discrete Geometry Extremal Combinatorics Turán-type Problems

Education

I completed my bachelor's degrees in Mathematics and Computer Science at KIT (Karlsruhe Institute of Technology) in Karlsruhe.

My bachelor thesis “Induced Turán problems” was supervised by Maria Axenovich and Thorsten Ueckerdt. The thesis studies extremal problems for induced and biinduced subgraphs, connections to Vapnik–Chervonenkis dimension, and includes a simplified proof of the Erdős–Hajnal conjecture for graphs with bounded VC dimension.