The signature kernel is a kernel between time series of arbitrary length and comes with strong theoretical guarantees from stochastic analysis. It has found applications in machine learning such as covariance functions for Gaussian processes. A strength of the underlying signature features is that they provide a structured global description of a time series. However, this property can quickly become a curse when local information is essential and forgetting is required; so far this has only been addressed with ad-hoc methods such as slicing the time series into smaller segments. To overcome this, we propose a principled and data-driven approach by introducing a novel forgetting mechanism for signature features. This allows the model to dynamically adapt its observed context length to focus on more recent information. To achieve this, we revisit the recently introduced Random Fourier Signature Features, and develop Random Fourier Decayed Signature Features (RFDSF) with Gaussian processes (GPs). This results in a Bayesian time series forecasting algorithm with variational inference, that offers a scalable probabilistic algorithm that processes and transforms a time series into a joint predictive distribution over the time steps in one pass using recurrence. For example, processing a sequence of length $10^4$ steps in less than $10^{-2}$ seconds and in $\approx$ 1GB of GPU memory. We demonstrate that the algorithm outperforms other GP-based alternatives and competes with state-of-the-art probabilistic time series forecasting algorithms.

Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves according to a diffusion equation defined using the graph Laplacian. We extend this approach by leveraging classic mathematical results about hypo-elliptic diffusions. This results in a novel tensor-valued graph operator, which we call the hypo-elliptic graph Laplacian. We provide theoretical guarantees and efficient low-rank approximation algorithms. In particular, this gives a structured approach to capture long-range dependencies on graphs that is robust to pooling. Besides the attractive theoretical properties, our experiments show that this method competes with graph transformers on datasets requiring long-range reasoning but scales only linearly in the number of edges as opposed to quadratically in nodes.

We develop a Bayesian approach to learning from sequential data by using Gaussian processes (GPs) with so-called signature kernels as covariance functions. This allows to make sequences of different length comparable and to rely on strong theoretical results from stochastic analysis. Signatures capture sequential structure with tensors that can scale unfavourably in sequence length and state space dimension. To deal with this, we introduce a sparse variational approach with inducing tensors. We then combine the resulting GP with LSTMs and GRUs to build larger models that leverage the strengths of each of these approaches and benchmark the resulting GPs on multivariate time series (TS) classification datasets.

), however, when databases are maintained by disparate organizations, the disclosure of such information can breach the privacy of the corresponding individuals. Various private record linkage (PRL) methods have been developed to obscure such identifiers, but they vary widely in their ability to balance competing goals of accuracy, efficiency and security. The tokenization and hashing of field values into Bloom filters (BF) enables greater linkage accuracy and efficiency than other PRL methods, but the encodings may be compromised through frequency-based cryptanalysis. Our objective is to adapt a BF encoding technique to mitigate such attacks with minimal sacrifices in accuracy and efficiency. To accomplish these goals, we introduce a statistically-informed method to generate BF encodings that integrate bits from multiple fields, the frequencies of which are provably associated with a minimum number of fields. Our method enables a user-specified tradeoff between security and accuracy. We compare our encoding method with other techniques using a public dataset of voter registration records and demonstrate that the increases in security come with only minor losses to accuracy.

OBJECTIVE: Integration of patients' records across resources enhances analytics. To address privacy concerns, emerging strategies such as Bloom filter encodings (BFEs), enable integration while obscuring identifiers. However, recent investigations demonstrate BFEs are, in theory, vulnerable to cryptanalysis when encoded identifiers are randomly selected from a public resource. This study investigates the extent to which cryptanalysis conditions hold for (1) real patient records and (2) a countermeasure that obscures the frequencies of the identifying values in encoded datasets. DESIGN: First, to investigate the strength of cryptanalysis for real patient records, we build BFEs from identifiers in an electronic medical record system and apply cryptanalysis using identifiers in a publicly available voter registry. Second, to investigate the countermeasure under ideal cryptanalysis conditions, we compose BFEs from the identifiers that are randomly selected from a public voter registry. MEASUREMENT: We utilize precision (ie, rate of correct re-identified encodings) and computation efficiency (ie, time to complete cryptanalysis) to assess the performance of cryptanalysis in BFEs before and after application of the countermeasure. RESULTS: Cryptanalysis can achieve high precision when the encoded identifiers are composed of a random sample of a public resource (ie, a voter registry). However, we also find that the attack is less efficient and may not be practical for more realistic scenarios. By contrast, the proposed countermeasure made cryptanalysis impractical in terms of precision and efficiency. CONCLUSIONS: Performance of cryptanalysis against BFEs based on patient data is significantly lower than theoretical estimates. The proposed countermeasure makes BFEs resistant to known practical attacks.