Robert Peharz

TMLR Journal 2025 Journal Article

What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?

• Lorenzo Loconte
• Antonio Mari
• Gennaro Gala
• Robert Peharz
• Cassio de Campos
• Erik Quaeghebeur
• Gennaro Vessio
• Antonio Vergari

This paper establishes a rigorous connection between circuit representations and tensor factorizations, two seemingly distinct yet fundamentally related areas. By connecting these fields, we highlight a series of opportunities that can benefit both communities. Our work generalizes popular tensor factorizations within the circuit language, and unifies various circuit learning algorithms under a single, generalized hierarchical factorization framework. Specifically, we introduce a modular “Lego block” approach to build tensorized circuit architectures. This, in turn, allows us to systematically construct and explore various circuit and tensor factorization models while maintaining tractability. This connection not only clarifies similarities and differences in existing models, but also enables the development of a comprehensive pipeline for building and optimizing new circuit/tensor factorization architectures. We show the effectiveness of our framework through extensive empirical evaluations, and highlight new research opportunities for tensor factorizations in probabilistic modeling.

ICML Conference 2024 Conference Paper

Exact Soft Analytical Side-Channel Attacks using Tractable Circuits

• Thomas Wedenig
• Rishub Nagpal
• Gaëtan Cassiers
• Stefan Mangard
• Robert Peharz

Detecting weaknesses in cryptographic algorithms is of utmost importance for designing secure information systems. The state-of-the-art soft analytical side-channel attack (SASCA) uses physical leakage information to make probabilistic predictions about intermediate computations and combines these "guesses" with the known algorithmic logic to compute the posterior distribution over the key. This attack is commonly performed via loopy belief propagation, which, however, lacks guarantees in terms of convergence and inference quality. In this paper, we develop a fast and exact inference method for SASCA, denoted as ExSASCA, by leveraging knowledge compilation and tractable probabilistic circuits. When attacking the Advanced Encryption Standard (AES), the most widely used encryption algorithm to date, ExSASCA outperforms SASCA by more than 31% top-1 success rate absolute. By leveraging sparse belief messages, this performance is achieved with little more computational cost than SASCA, and about 3 orders of magnitude less than exact inference via exhaustive enumeration. Even with dense belief messages, ExSASCA still uses 6 times less computations than exhaustive inference.

JMLR Journal 2024 Journal Article

Resource-Efficient Neural Networks for Embedded Systems

• Wolfgang Roth
• Günther Schindler
• Bernhard Klein
• Robert Peharz
• Sebastian Tschiatschek
• Holger Fröning
• Franz Pernkopf
• Zoubin Ghahramani

While machine learning is traditionally a resource intensive task, embedded systems, autonomous navigation, and the vision of the Internet of Things fuel the interest in resource-efficient approaches. These approaches aim for a carefully chosen trade-off between performance and resource consumption in terms of computation and energy. The development of such approaches is among the major challenges in current machine learning research and key to ensure a smooth transition of machine learning technology from a scientific environment with virtually unlimited computing resources into everyday's applications. In this article, we provide an overview of the current state of the art of machine learning techniques facilitating these real-world requirements. In particular, we focus on resource-efficient inference based on deep neural networks (DNNs), the predominant machine learning models of the past decade. We give a comprehensive overview of the vast literature that can be mainly split into three non-mutually exclusive categories: (i) quantized neural networks, (ii) network pruning, and (iii) structural efficiency. These techniques can be applied during training or as post-processing, and they are widely used to reduce the computational demands in terms of memory footprint, inference speed, and energy efficiency. We also briefly discuss different concepts of embedded hardware for DNNs and their compatibility with machine learning techniques as well as potential for energy and latency reduction. We substantiate our discussion with experiments on well-known benchmark data sets using compression techniques (quantization, pruning) for a set of resource-constrained embedded systems, such as CPUs, GPUs and FPGAs. The obtained results highlight the difficulty of finding good trade-offs between resource efficiency and prediction quality. [abs] [ pdf ][ bib ] &copy JMLR 2024. ( edit, beta )

AAAI Conference 2023 Conference Paper

Continuous Mixtures of Tractable Probabilistic Models

• Alvaro H.C. Correia
• Gennaro Gala
• Erik Quaeghebeur
• Cassio de Campos
• Robert Peharz

Probabilistic models based on continuous latent spaces, such as variational autoencoders, can be understood as uncountable mixture models where components depend continuously on the latent code. They have proven to be expressive tools for generative and probabilistic modelling, but are at odds with tractable probabilistic inference, that is, computing marginals and conditionals of the represented probability distribution. Meanwhile, tractable probabilistic models such as probabilistic circuits (PCs) can be understood as hierarchical discrete mixture models, and thus are capable of performing exact inference efficiently but often show subpar performance in comparison to continuous latent-space models. In this paper, we investigate a hybrid approach, namely continuous mixtures of tractable models with a small latent dimension. While these models are analytically intractable, they are well amenable to numerical integration schemes based on a finite set of integration points. With a large enough number of integration points the approximation becomes de-facto exact. Moreover, for a finite set of integration points, the integration method effectively compiles the continuous mixture into a standard PC. In experiments, we show that this simple scheme proves remarkably effective, as PCs learnt this way set new state of the art for tractable models on many standard density estimation benchmarks.

NeurIPS Conference 2023 Conference Paper

How to Turn Your Knowledge Graph Embeddings into Generative Models

• Lorenzo Loconte
• Nicola Di Mauro
• Robert Peharz
• Antonio Vergari

Some of the most successful knowledge graph embedding (KGE) models for link prediction – CP, RESCAL, TuckER, ComplEx – can be interpreted as energy-based models. Under this perspective they are not amenable for exact maximum-likelihood estimation (MLE), sampling and struggle to integrate logical constraints. This work re-interprets the score functions of these KGEs as circuits – constrained computational graphs allowing efficient marginalisation. Then, we design two recipes to obtain efficient generative circuit models by either restricting their activations to be non-negative or squaring their outputs. Our interpretation comes with little or no loss of performance for link prediction, while the circuits framework unlocks exact learning by MLE, efficient sampling of new triples, and guarantee that logical constraints are satisfied by design. Furthermore, our models scale more gracefully than the original KGEs on graphs with millions of entities.

NeurIPS Conference 2022 Conference Paper

Active Bayesian Causal Inference

• Christian Toth
• Lars Lorch
• Christian Knoll
• Andreas Krause
• Franz Pernkopf
• Robert Peharz
• Julius von Kügelgen

Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e. g. , the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference—quantities that are not of direct interest ought to be marginalized out in this process, thus contributing to our overall uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, i. e. , for jointly inferring a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient nonlinear additive Gaussian noise models, which we model using Gaussian processes. To capture the space of causal graphs, we use a continuous latent graph representation, allowing our approach to scale to practically relevant problem sizes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, update our beliefs, and repeat. Through simulations, we demonstrate that our approach is more data-efficient than existing methods that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples, while providing well-calibrated uncertainty estimates of the quantities of interest.

ICML Conference 2020 Conference Paper

Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits

• Robert Peharz
• Steven Lang
• Antonio Vergari
• Karl Stelzner
• Alejandro Molina 0001
• Martin Trapp 0001
• Guy Van den Broeck
• Kristian Kersting

Probabilistic circuits (PCs) are a promising avenue for probabilistic modeling, as they permit a wide range of exact and efficient inference routines. Recent “deep-learning-style” implementations of PCs strive for a better scalability, but are still difficult to train on real-world data, due to their sparsely connected computational graphs. In this paper, we propose Einsum Networks (EiNets), a novel implementation design for PCs, improving prior art in several regards. At their core, EiNets combine a large number of arithmetic operations in a single monolithic einsum-operation, leading to speedups and memory savings of up to two orders of magnitude, in comparison to previous implementations. As an algorithmic contribution, we show that the implementation of Expectation-Maximization (EM) can be simplified for PCs, by leveraging automatic differentiation. Furthermore, we demonstrate that EiNets scale well to datasets which were previously out of reach, such as SVHN and CelebA, and that they can be used as faithful generative image models.

NeurIPS Conference 2020 Conference Paper

Joints in Random Forests

• Alvaro Correia
• Robert Peharz
• Cassio P. de Campos

Decision Trees (DTs) and Random Forests (RFs) are powerful discriminative learners and tools of central importance to the everyday machine learning practitioner and data scientist. Due to their discriminative nature, however, they lack principled methods to process inputs with missing features or to detect outliers, which requires pairing them with imputation techniques or a separate generative model. In this paper, we demonstrate that DTs and RFs can naturally be interpreted as generative models, by drawing a connection to Probabilistic Circuits, a prominent class of tractable probabilistic models. This reinterpretation equips them with a full joint distribution over the feature space and leads to Generative Decision Trees (GeDTs) and Generative Forests (GeFs), a family of novel hybrid generative-discriminative models. This family of models retains the overall characteristics of DTs and RFs while additionally being able to handle missing features by means of marginalisation. Under certain assumptions, frequently made for Bayes consistency results, we show that consistency in GeDTs and GeFs extend to any pattern of missing input features, if missing at random. Empirically, we show that our models often outperform common routines to treat missing data, such as K-nearest neighbour imputation, and moreover, that our models can naturally detect outliers by monitoring the marginal probability of input features.

AAAI Conference 2019 Conference Paper

Automatic Bayesian Density Analysis

• Antonio Vergari
• Alejandro Molina
• Robert Peharz
• Zoubin Ghahramani
• Kristian Kersting
• Isabel Valera

Making sense of a dataset in an automatic and unsupervised fashion is a challenging problem in statistics and AI. Classical approaches for exploratory data analysis are usually not flexible enough to deal with the uncertainty inherent to real-world data: they are often restricted to fixed latent interaction models and homogeneous likelihoods; they are sensitive to missing, corrupt and anomalous data; moreover, their expressiveness generally comes at the price of intractable inference. As a result, supervision from statisticians is usually needed to find the right model for the data. However, since domain experts are not necessarily also experts in statistics, we propose Automatic Bayesian Density Analysis (ABDA) to make exploratory data analysis accessible at large. Specifically, ABDA allows for automatic and efficient missing value estimation, statistical data type and likelihood discovery, anomaly detection and dependency structure mining, on top of providing accurate density estimation. Extensive empirical evidence shows that ABDA is a suitable tool for automatic exploratory analysis of mixed continuous and discrete tabular data.

NeurIPS Conference 2019 Conference Paper

Bayesian Learning of Sum-Product Networks

• Martin Trapp
• Robert Peharz
• Hong Ge
• Franz Pernkopf
• Zoubin Ghahramani

Sum-product networks (SPNs) are flexible density estimators and have received significant attention due to their attractive inference properties. While parameter learning in SPNs is well developed, structure learning leaves something to be desired: Even though there is a plethora of SPN structure learners, most of them are somewhat ad-hoc and based on intuition rather than a clear learning principle. In this paper, we introduce a well-principled Bayesian framework for SPN structure learning. First, we decompose the problem into i) laying out a computational graph, and ii) learning the so-called scope function over the graph. The first is rather unproblematic and akin to neural network architecture validation. The second represents the effective structure of the SPN and needs to respect the usual structural constraints in SPN, i. e. completeness and decomposability. While representing and learning the scope function is somewhat involved in general, in this paper, we propose a natural parametrisation for an important and widely used special case of SPNs. These structural parameters are incorporated into a Bayesian model, such that simultaneous structure and parameter learning is cast into monolithic Bayesian posterior inference. In various experiments, our Bayesian SPNs often improve test likelihoods over greedy SPN learners. Further, since the Bayesian framework protects against overfitting, we can evaluate hyper-parameters directly on the Bayesian model score, waiving the need for a separate validation set, which is especially beneficial in low data regimes. Bayesian SPNs can be applied to heterogeneous domains and can easily be extended to nonparametric formulations. Moreover, our Bayesian approach is the first, which consistently and robustly learns SPN structures under missing data.

ICML Conference 2019 Conference Paper

Faster Attend-Infer-Repeat with Tractable Probabilistic Models

• Karl Stelzner
• Robert Peharz
• Kristian Kersting

The recent Attend-Infer-Repeat (AIR) framework marks a milestone in structured probabilistic modeling, as it tackles the challenging problem of unsupervised scene understanding via Bayesian inference. AIR expresses the composition of visual scenes from individual objects, and uses variational autoencoders to model the appearance of those objects. However, inference in the overall model is highly intractable, which hampers its learning speed and makes it prone to suboptimal solutions. In this paper, we show that the speed and robustness of learning in AIR can be considerably improved by replacing the intractable object representations with tractable probabilistic models. In particular, we opt for sum-product networks (SPNs), expressive deep probabilistic models with a rich set of tractable inference routines. The resulting model, called SuPAIR, learns an order of magnitude faster than AIR, treats object occlusions in a consistent manner, and allows for the inclusion of a background noise model, improving the robustness of Bayesian scene understanding.

ICML Conference 2019 Conference Paper

Hierarchical Decompositional Mixtures of Variational Autoencoders

• Ping Liang Tan
• Robert Peharz

Variational autoencoders (VAEs) have received considerable attention, since they allow us to learn expressive neural density estimators effectively and efficiently. However, learning and inference in VAEs is still problematic due to the sensitive interplay between the generative model and the inference network. Since these problems become generally more severe in high dimensions, we propose a novel hierarchical mixture model over low-dimensional VAE experts. Our model decomposes the overall learning problem into many smaller problems, which are coordinated by the hierarchical mixture, represented by a sum-product network. In experiments we show that our models outperform classical VAEs on almost all of our experimental benchmarks. Moreover, we show that our model is highly data efficient and degrades very gracefully in extremely low data regimes. ow data regimes.

UAI Conference 2019 Conference Paper

Random Sum-Product Networks: A Simple and Effective Approach to Probabilistic Deep Learning

• Robert Peharz
• Antonio Vergari
• Karl Stelzner
• Alejandro Molina 0001
• Martin Trapp 0001
• Xiaoting Shao
• Kristian Kersting
• Zoubin Ghahramani

Sum-product networks (SPNs) are expressive probabilistic models with a rich set of exact and efficient inference routines. However, in order to guarantee exact inference, they require specific structural constraints, which complicate learning SPNs from data. Thereby, most SPN structure learners proposed so far are tedious to tune, do not scale easily, and are not easily integrated with deep learning frameworks. In this paper, we follow a simple “deep learning” approach, by generating unspecialized random structures, scalable to millions of parameters, and subsequently applying GPU-based optimization. Somewhat surprisingly, our models often perform on par with state-of-the-art SPN structure learners and deep neural networks on a diverse range of generative and discriminative scenarios. At the same time, our models yield well-calibrated uncertainties, and stand out among most deep generative and discriminative models in being robust to missing features and being able to detect anomalies.

AAAI Conference 2018 Conference Paper

Sum-Product Autoencoding: Encoding and Decoding Representations Using Sum-Product Networks

• Antonio Vergari
• Robert Peharz
• Nicola Di Mauro
• Alejandro Molina
• Kristian Kersting
• Floriana Esposito

Sum-Product Networks (SPNs) are a deep probabilistic architecture that up to now has been successfully employed for tractable inference. Here, we extend their scope towards unsupervised representation learning: we encode samples into continuous and categorical embeddings and show that they can also be decoded back into the original input space by leveraging MPE inference. We characterize when this Sum- Product Autoencoding (SPAE) leads to equivalent reconstructions and extend it towards dealing with missing embedding information. Our experimental results on several multilabel classification problems demonstrate that SPAE is competitive with state-of-the-art autoencoder architectures, even if the SPNs were never trained to reconstruct their inputs.

UAI Conference 2017 Conference Paper

Safe Semi-Supervised Learning of Sum-Product Networks

• Martin Trapp 0001
• Tamas Madl
• Robert Peharz
• Franz Pernkopf
• Robert Trappl

In several domains obtaining class annotations is expensive while at the same time unlabelled data are abundant. While most semi-supervised approaches enforce restrictive assumptions on the data distribution, recent work has managed to learn semi-supervised models in a nonrestrictive regime. However, so far such approaches have only been proposed for linear models. In this work, we introduce semisupervised parameter learning for Sum-Product Networks (SPNs). SPNs are deep probabilistic models admitting inference in linear time in number of network edges. Our approach has several advantages, as it (1) allows generative and discriminative semi-supervised learning, (2) guarantees that adding unlabelled data can increase, but not degrade, the performance (safe), and (3) is computationally efficient and does not enforce restrictive assumptions on the data distribution. We show on a variety of data sets that safe semi-supervised learning with SPNs is competitive compared to state-of-theart and can lead to a better generative and discriminative objective value than a purely supervised approach.

ICML Conference 2013 Conference Paper

The Most Generative Maximum Margin Bayesian Networks

• Robert Peharz
• Sebastian Tschiatschek
• Franz Pernkopf

Although discriminative learning in graphical models generally improves classification results, the generative semantics of the model are compromised. In this paper, we introduce a novel approach of hybrid generative-discriminative learning for Bayesian networks. We use an SVM-type large margin formulation for discriminative training, introducing a likelihood-weighted \ell^1-norm for the SVM-norm-penalization. This simultaneously optimizes the data likelihood and therefore partly maintains the generative character of the model. For many network structures, our method can be formulated as a convex problem, guaranteeing a globally optimal solution. In terms of classification, the resulting models outperform state-of-the art generative and discriminative learning methods for Bayesian networks, and are comparable with linear and kernelized SVMs. Furthermore, the models achieve likelihoods close to the maximum likelihood solution and show robust behavior in classification experiments with missing features.

ICML Conference 2012 Conference Paper

Exact Maximum Margin Structure Learning of Bayesian Networks

• Robert Peharz
• Franz Pernkopf