arXiv subjects

Daniele Venturi

Publications and source records attributed to Daniele Venturi.

At least 19 recordsLinked to original sources

arXiv2026

Counting Residue-Class Survivor Sets: An Asymptotic Comparison with Prime-Admissible Sets

For $n\geq1$, choose one residue class $r_k\pmod{k}$ for every $2\leq k\leq n$, and retain those $m\in\{2,\ldots,n+1\}$ for which $m\not\equiv r_k\pmod{k}$ whenever $k<m$. Let $N(n)$ be the number of distinct survivor sets obtained in this way. We prove \[ \log 2\leq \liminf_{n\to\infty}\frac{\log N(n)}{n/\log n} \leq \limsup_{n\to\infty}\frac{\log N(n)}{n/\log n} \leq2\log 2. \] If $A_{\rm adm}(n)$ denotes the number of subsets of $\{2,\ldots,n+1\}$ that omit at least one residue class modulo every prime, then our main comparison is \[ \log N(n)=\log A_{\rm adm}(n)+o\!\left(\frac{n}{\log n}\right). \] We also obtain an asymptotic formula for the number of distinct intersections of survivor sets with the primes, and an exact recurrence $N(n+1)=N(n)+E(n)$, where $E(n)$ counts the survivor sets that can be extended by the new point $n+2$.

math.NT↗

arXiv2026

Holographic generative flows with AdS/CFT

Holography, in the form of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, offers a natural setting for generative modelling. Data on a boundary manifold lifts into a higher-dimensional bulk through a propagator, and this extra dimension plays the role of a flow parameter. We exploit this structure to build GenAdS, an approach to generative flow matching in which the dynamics are represented by the evolution of fields in AdS, together with a residual correction learned by a neural network. Boundary samples are encoded as scalar sources, transported into the bulk along the flow, and decoded after numerical integration. Our paradigm combines a Fourier-space encoding scheme for the data as AdS sources, a normalised radial phase space in which to stage the flow-matching dynamics, and a Klein--Gordon backbone to guide the flow. On a two-dimensional checkerboard benchmark, our experiments show that most of the benefit of GenAdS comes from the Fourier representation and convolutional architecture. However, when we remove momentum-channel regularisation, our most physics-informed GenAdS variant rivals the strongest physics-free control on boundary violation. On MNIST, GenAdS models remain close to a convolutional baseline on fidelity while achieving significantly higher recall at comparable precision, suggesting a fidelity-coverage trade-off. Our findings establish GenAdS as a physically interpretable and experimentally controllable framework for generative modelling, with many avenues for future extension.

cs.LG↗

arXiv2026

Uncertainty propagation in auto-regressive random neural network models

We develop analytical and particle-based methods for uncertainty propagation in random neural network models, where both the inputs and network parameters are allowed to be random. Building on the piecewise-linear structure of the Leaky ReLU activation function, we derive a local approximation of the neural network output with respect to perturbations in both its inputs and parameters. This approximation is exact for perturbations that preserve the network activation pattern, and it allows us to compute analytical expressions for the probability density function and characteristic function of the network output, together with closed-form approximations for its mean and covariance. We extend this uncertainty propagation framework to autonomous dynamical systems whose one-step evolution map is represented by a random neural network. Repeated application of this map defines an autoregressive model, for which we derive recursive equations to propagate uncertainty in both the state and network parameters over time. These equations explicitly account for the state-parameter cross-covariance that develops under successive iterations of the network. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate uncertainty propagation through the predictability horizon and the applicability of the proposed framework to high-dimensional dynamical systems.

stat.ML↗

arXiv2026

Computing with traceable tensor networks

We introduce a new SVD-based tensor decomposition method for tensor networks with arbitrary graph topologies, extending classical hierarchical SVD-based techniques to networks with cycles and general connectivity. We also introduce addition and rounding procedures for traceable tensor graphs, enabling step-rounding time integration of high-dimensional PDEs directly in graph format, with rank truncation controlled to a prescribed tolerance at every time step. We demonstrate the new method on the decomposition of multivariate functions and on the numerical solution of the Fokker-Planck equation, and find that the graph-format representation attains comparable or better accuracy than the classical tensor train and hierarchical Tucker tensor formats, while using substantially fewer degrees of freedom at lower computational cost.

physics.comp-ph↗

arXiv2026

Weighted Flow Matching and Physics-Informed Nonlinear Filtering for Parameter Estimation in Digital Twins

Digital twins (DTs) rely on continuous synchronization between physical systems and their virtual counterparts through online parameter estimation under uncertainty. In many practical settings, however, this task is challenged by low observability, weak excitation, nonlinear dynamics, and noisy or biased measurements. In this work, we develop a new mathematical framework that integrates Weighted Flow Matching (WFM) generative modeling with physics-informed nonlinear filtering to enhance parameter estimation in DTs. WFM relies on dynamic reweighting of training samples, which guides the generative model toward parameter regimes most informative of the evolving system state. This generative component is tightly coupled with a physics-informed filtering architecture based on the Unscented Kalman Filter (UKF), yielding a unified DT framework that combines data-driven probability transport with physically consistent state and parameter estimation. The effectiveness of the new integrated framework is demonstrated within a spacecraft DT architecture, where stable moment of inertia estimation is achieved under uncertain and noisy sensing, with significant performance improvements over established approaches such as Extended Kalman Filtering (EKF) and Ensemble Kalman Filtering (EnKF). These results highlight the potential of weighted generative modeling as a core mechanism for real-time DT synchronization in operational and mission-critical systems.

cs.CE↗

arXiv2026

The Coding Limits of Robust Watermarking for Generative Models

We study a basic question about cryptographic watermarking for generative models: how reliable can a watermark remain when an adversary is allowed to corrupt the encoded signal? To address this question, we introduce a minimal coding abstraction that we call a zero-bit tamper-detection code. This is a secret-key procedure that samples a pseudorandom codeword and, given a candidate word, decides whether it should be treated as unmarked content or as the result of tampering with a valid codeword. It captures the two core requirements of robust watermarking: soundness and tamper detection. Within this abstraction we prove a sharp unconditional limit on robustness to independent symbol corruption. For an alphabet of size $q$, there is a critical corruption rate of $1-1/q$ such that no scheme with soundness, even relaxed to allow a fixed constant false positive probability on random content, can reliably detect tampering once an adversary can change more than this fraction of symbols. In particular, in the binary case no cryptographic watermark can remain robust if more than half of the encoded bits are modified. We also show that this threshold is tight by giving simple information-theoretic constructions that achieve soundness and tamper detection for all strictly smaller corruption rates. We then test experimentally whether this limit appears in practice by looking at the recent watermarking for images of Gunn, Zhao, and Song (ICLR 2025). We show that a simple crop and resize operation reliably flipped about half of the latent signs and consistently prevented belief-propagation decoding from recovering the codeword, erasing the watermark while leaving the image visually intact.

cs.CR↗

arXiv2026

Score-Based Diffusion Models in Infinite Dimensions: A Malliavin Calculus Perspective

We study score-based diffusion modelling in infinite-dimensional separable Hilbert spaces through Malliavin calculus, extending the analysis of generative models beyond the finite-dimensional setting. The forward diffusion process is formulated as a linear stochastic partial differential equation (SPDE) driven by space--time coloured noise with a trace-class covariance operator, ensuring well-posedness in arbitrary spatial dimensions. Building on Malliavin calculus and an infinite-dimensional extension of the Bismut--Elworthy--Li formula, we derive a closed-form expression for the logarithmic derivative of the transition measure along Cameron--Martin directions, which serves as the natural infinite-dimensional analogue of the score function. Our operator-theoretic approach preserves the intrinsic geometry of Hilbert spaces and accommodates general trace-class operators, thereby incorporating spatially correlated noise without assuming semigroup invertibility. We validate the derived score formula numerically for several classes of linear SPDEs in both one and two spatial dimensions using spectral methods.

math.PR↗

arXiv2025

Generative forecasting with joint probability models

Chaotic dynamical systems exhibit strong sensitivity to initial conditions and often contain unresolved multiscale processes, making deterministic forecasting fundamentally limited. Generative models offer an appealing alternative by learning distributions over plausible system evolutions; yet, most existing approaches focus on next-step conditional prediction rather than the structure of the underlying dynamics. In this work, we reframe forecasting as a fully generative problem by learning the joint probability distribution of lagged system states over short temporal windows and obtaining forecasts through marginalization. This new perspective allows the model to capture nonlinear temporal dependencies, represent multistep trajectory segments, and produce next-step predictions consistent with the learned joint distribution. We also introduce a general, model-agnostic training and inference framework for joint generative forecasting and show how it enables assessment of forecast robustness and reliability using three complementary uncertainty quantification metrics (ensemble variance, short-horizon autocorrelation, and cumulative Wasserstein drift), without access to ground truth. We evaluate the performance of the proposed method on two canonical chaotic dynamical systems, the Lorenz-63 system and the Kuramoto-Sivashinsky equation, and show that joint generative models yield improved short-term predictive skill, preserve attractor geometry, and achieve substantially more accurate long-range statistical behaviour than conventional conditional next-step models.

cs.LG↗

arXiv2025

Malliavin Calculus for Score-based Diffusion Models

We introduce a new framework based on Malliavin calculus to derive exact analytical expressions for the score function $\nabla \log p_t(x)$, i.e., the gradient of the log-density associated with the solution to stochastic differential equations (SDEs). Our approach combines classical integration-by-parts techniques with modern stochastic analysis tools, such as Bismut's formula and Malliavin calculus, and it works for both linear and nonlinear SDEs. In doing so, we establish a rigorous connection between the Malliavin derivative, its adjoint, the Malliavin divergence (Skorokhod integral), and diffusion generative models, thereby providing a systematic method for computing $\nabla \log p_t(x)$. In the linear case, we present a detailed analysis showing that our formula coincides with the analytical score function derived from the solution of the Fokker--Planck equation. For nonlinear SDEs with state-independent diffusion coefficients, we derive a closed-form expression for $\nabla \log p_t(x)$. We evaluate the proposed framework across multiple generative tasks and find that its performance is comparable to state-of-the-art methods. These results can be generalised to broader classes of SDEs, paving the way for new score-based diffusion generative models.

cs.LG↗

arXiv2025

A Malliavin calculus approach to score functions in diffusion generative models

Score-based diffusion generative models have recently emerged as a powerful tool for modelling complex data distributions. These models aim at learning the score function, which defines a map from a known probability distribution to the target data distribution via deterministic or stochastic differential equations (SDEs). The score function is typically estimated from data using a variety of approximation techniques, such as denoising or sliced score matching, Hyvärien's method, or Schrödinger bridges. In this paper, we derive an exact, closed-form, expression for the score function for a broad class of nonlinear diffusion generative models. Our approach combines modern stochastic analysis tools such as Malliavin derivatives and their adjoint operators (Skorokhod integrals or Malliavin Divergence) with a new Bismut-type formula. The resulting expression for the score function can be written entirely in terms of the first and second variation processes, with all Malliavin derivatives systematically eliminated, thereby enhancing its practical applicability. The theoretical framework presented in this work offers a principled foundation for advancing score estimation methods in generative modelling, enabling the design of new sampling algorithms for complex probability distributions. Our results can be extended to broader classes of stochastic differential equations, opening new directions for the development of score-based diffusion generative models.

stat.ML↗

arXiv2025

Uncertainty propagation in feed-forward neural network models

We develop new uncertainty propagation methods for feed-forward neural network architectures with leaky ReLU activation functions subject to random perturbations in the input vectors. In particular, we derive analytical expressions for the probability density function (PDF) of the neural network output and its statistical moments as a function of the input uncertainty and the parameters of the network, i.e., weights and biases. A key finding is that an appropriate linearization of the leaky ReLU activation function yields accurate statistical results even for large perturbations in the input vectors. This can be attributed to the way information propagates through the network. We also propose new analytically tractable Gaussian copula surrogate models to approximate the full joint PDF of the neural network output. To validate our theoretical results, we conduct Monte Carlo simulations and a thorough error analysis on a multi-layer neural network representing a nonlinear integro-differential operator between two polynomial function spaces. Our findings demonstrate excellent agreement between the theoretical predictions and Monte Carlo simulations.

cs.LG↗

arXiv2025

MARTSIA: A Tool for Confidential Data Exchange via Public Blockchain

Blockchain technology streamlines multi-party collaborations in decentralized settings, especially when trust is limited or difficult to establish. While public blockchains enhance transparency and reliability by replicating data across all network nodes, they also conflict with confidentiality. Here, we introduce Multi-Authority Approach to Transaction Systems for Interoperating Applications (MARTSIA) to address this challenge. MARTSIA provides fine-grained read-access control at the message-part level by combining user-defined policies with certifier-declared attributes. The approach guarantees that even though data is replicated across the network to maintain consistency, fault tolerance, and availability, its confidentiality is securely preserved through encryption. To this end, MARTSIA integrates blockchain technologies, Multi-Authority Attribute-Based Encryption, and distributed hash-table file storages. This architecture effectively balances the transparency inherent in public blockchains with the privacy required for sensitive applications. We present the tool and its applicability in a business scenario.

cs.CR↗

arXiv2025

Watermarks in the Sand: Impossibility of Strong Watermarking for Generative Models

Watermarking generative models consists of planting a statistical signal (watermark) in a model's output so that it can be later verified that the output was generated by the given model. A strong watermarking scheme satisfies the property that a computationally bounded attacker cannot erase the watermark without causing significant quality degradation. In this paper, we study the (im)possibility of strong watermarking schemes. We prove that, under well-specified and natural assumptions, strong watermarking is impossible to achieve. This holds even in the private detection algorithm setting, where the watermark insertion and detection algorithms share a secret key, unknown to the attacker. To prove this result, we introduce a generic efficient watermark attack; the attacker is not required to know the private key of the scheme or even which scheme is used. Our attack is based on two assumptions: (1) The attacker has access to a "quality oracle" that can evaluate whether a candidate output is a high-quality response to a prompt, and (2) The attacker has access to a "perturbation oracle" which can modify an output with a nontrivial probability of maintaining quality, and which induces an efficiently mixing random walk on high-quality outputs. We argue that both assumptions can be satisfied in practice by an attacker with weaker computational capabilities than the watermarked model itself, to which the attacker has only black-box access. Furthermore, our assumptions will likely only be easier to satisfy over time as models grow in capabilities and modalities. We demonstrate the feasibility of our attack by instantiating it to attack three existing watermarking schemes for large language models: Kirchenbauer et al. (2023), Kuditipudi et al. (2023), and Zhao et al. (2023). The same attack successfully removes the watermarks planted by all three schemes, with only minor quality degradation.

cs.LG↗

arXiv2024

Enabling Data Confidentiality with Public Blockchains

Blockchain technology is apt to facilitate the automation of multi-party cooperations among various players in a decentralized setting, especially in cases where trust among participants is limited. Transactions are stored in a ledger, a replica of which is retained by every node of the blockchain network. The operations saved thereby are thus publicly accessible. While this aspect enhances transparency, reliability, and persistence, it hinders the utilization of public blockchains for process automation as it violates typical confidentiality requirements in corporate settings. To overcome this issue, we propose our approach named Multi-Authority Approach to Transaction Systems for Interoperating Applications (MARTSIA). Based on Multi-Authority Attribute-Based Encryption (MA-ABE), MARTSIA enables read-access control over shared data at the level of message parts. User-defined policies determine whether an actor can interpret the publicly stored information or not, depending on the actor's attributes declared by a consortium of certifiers. Still, all nodes in the blockchain network can attest to the publication of the (encrypted) data. We provide a formal analysis of the security guarantees of MARTSIA, and illustrate the proof-of-concept implementation over multiple blockchain platforms. To demonstrate its interoperability, we showcase its usage in ensemble with a state-of-the-art blockchain-based engine for multi-party process execution, and three real-world decentralized applications in the context of NFT markets, supply chain, and retail.

cs.CR↗

arXiv2024

Tensor approximation of functional differential equations

Functional Differential Equations (FDEs) play a fundamental role in many areas of mathematical physics, including fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equation), and statistical physics. Despite their significance, computing solutions to FDEs remains a longstanding challenge in mathematical physics. In this paper we address this challenge by introducing new approximation theory and high-performance computational algorithms designed for solving FDEs on tensor manifolds. Our approach involves approximating FDEs using high-dimensional partial differential equations (PDEs), and then solving such high-dimensional PDEs on a low-rank tensor manifold leveraging high-performance parallel tensor algorithms. The effectiveness of the proposed approach is demonstrated through its application to the Burgers-Hopf FDE, which governs the characteristic functional of the stochastic solution to the Burgers equation evolving from a random initial state.

math.NA↗

arXiv2023

MARTSIA: Enabling Data Confidentiality for Blockchain-based Process Execution

Multi-party business processes rely on the collaboration of various players in a decentralized setting. Blockchain technology can facilitate the automation of these processes, even in cases where trust among participants is limited. Transactions are stored in a ledger, a replica of which is retained by every node of the blockchain network. The operations saved thereby are thus publicly accessible. While this enhances transparency, reliability, and persistence, it hinders the utilization of public blockchains for process automation as it violates typical confidentiality requirements in corporate settings. In this paper, we propose MARTSIA: A Multi-Authority Approach to Transaction Systems for Interoperating Applications. MARTSIA enables precise control over process data at the level of message parts. Based on Multi-Authority Attribute-Based Encryption (MA-ABE), MARTSIA realizes a number of desirable properties, including confidentiality, transparency, and auditability. We implemented our approach in proof-of-concept prototypes, with which we conduct a case study in the area of supply chain management. Also, we show the integration of MARTSIA with a state-of-the-art blockchain-based process execution engine to secure the data flow.

cs.CR↗

arXiv2023

Coordinate-adaptive integration of PDEs on tensor manifolds

We introduce a new tensor integration method for time-dependent PDEs that controls the tensor rank of the PDE solution via time-dependent diffeomorphic coordinate transformations. Such coordinate transformations are generated by minimizing the normal component of the PDE operator relative to the tensor manifold that approximates the PDE solution via a convex functional. The proposed method significantly improves upon and may be used in conjunction with the coordinate-adaptive algorithm we recently proposed in JCP (2023) Vol. 491, 112378, which is based on non-convex relaxations of the rank minimization problem and Riemannian optimization. Numerical applications demonstrating the effectiveness of the proposed coordinate-adaptive tensor integration method are presented and discussed for prototype Liouville and Fokker-Planck equations.

math.NA↗

arXiv2023

The Mori-Zwanzig formulation of deep learning

We develop a new formulation of deep learning based on the Mori-Zwanzig (MZ) formalism of irreversible statistical mechanics. The new formulation is built upon the well-known duality between deep neural networks and discrete dynamical systems, and it allows us to directly propagate quantities of interest (conditional expectations and probability density functions) forward and backward through the network by means of exact linear operator equations. Such new equations can be used as a starting point to develop new effective parameterizations of deep neural networks, and provide a new framework to study deep-learning via operator theoretic methods. The proposed MZ formulation of deep learning naturally introduces a new concept, i.e., the memory of the neural network, which plays a fundamental role in low-dimensional modeling and parameterization. By using the theory of contraction mappings, we develop sufficient conditions for the memory of the neural network to decay with the number of layers. This allows us to rigorously transform deep networks into shallow ones, e.g., by reducing the number of neurons per layer (using projection operators), or by reducing the total number of layers (using the decay property of the memory operator).

cs.LG↗