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Nutrient load simulators are large, deterministic, models that simulate the hydrodynamics and biogeochemical processes in aquatic ecosystems. They are central tools for planning cost efficient actions to fight eutrophication since they allow scenario predictions on impacts of nutrient load reductions to, e.g., harmful algal biomass growth. Due to being computationally heavy, the uncertainties related to these predictions are typically not rigorously assessed though. In this work, we developed a novel Bayesian computational approach for estimating the uncertainties in predictions of the Finnish coastal nutrient load model FICOS. First, we constructed a likelihood function for the multivariate spatiotemporal outputs of the FICOS model. Then, we used Bayes optimization to locate the posterior mode for the model parameters conditional on long term monitoring data. After that, we constructed a space filling design for FICOS model runs around the posterior mode and used it to train a Gaussian process emulator for the (log) posterior density of the model parameters. We then integrated over this (approximate) parameter posterior to produce probabilistic predictions for algal biomass and chlorophyll a concentration under alternative nutrient load reduction scenarios. Our computational algorithm allowed for fast posterior inference and the Gaussian process emulator had good predictive accuracy within the highest posterior probability mass region. The posterior predictive scenarios showed that the probability to reach the EUs Water Framework Directive objectives in the Finnish Archipelago Sea is generally low even under large load reductions.

We present a constructive universal approximation theorem for learning machines equipped with joint-group-equivariant feature maps, based on the group representation theory. ``Constructive'' here indicates that the distribution of parameters is given in a closed-form expression known as the ridgelet transform. Joint-group-equivariance encompasses a broad class of feature maps that generalize classical group-equivariance. Notably, this class includes fully-connected networks, which are not group-equivariant but are joint-group-equivariant. Moreover, our main theorem also unifies the universal approximation theorems for both shallow and deep networks. While the universality of shallow networks has been investigated in a unified manner by the ridgelet transform, the universality of deep networks has been investigated in a case-by-case manner.

State space models (SSMs) leverage linear, time-invariant (LTI) systems to effectively learn sequences with long-range dependencies. By analyzing the transfer functions of LTI systems, we find that SSMs exhibit an implicit bias toward capturing low-frequency components more effectively than high-frequency ones. This behavior aligns with the broader notion of frequency bias in deep learning model training. We show that the initialization of an SSM assigns it an innate frequency bias and that training the model in a conventional way does not alter this bias. Based on our theory, we propose two mechanisms to tune frequency bias: either by scaling the initialization to tune the inborn frequency bias; or by applying a Sobolev-norm-based filter to adjust the sensitivity of the gradients to high-frequency inputs, which allows us to change the frequency bias via training. Using an image-denoising task, we empirically show that we can strengthen, weaken, or even reverse the frequency bias using both mechanisms. By tuning the frequency bias, we can also improve SSMs' performance on learning long-range sequences, averaging an 88.26% accuracy on the Long-Range Arena (LRA) benchmark tasks.

Probabilistic relational models provide a well-established formalism to combine first-order logic and probabilistic models, thereby allowing to represent relationships between objects in a relational domain. At the same time, the field of artificial intelligence requires increasingly large amounts of relational training data for various machine learning tasks. Collecting real-world data, however, is often challenging due to privacy concerns, data protection regulations, high costs, and so on. To mitigate these challenges, the generation of synthetic data is a promising approach. In this paper, we solve the problem of generating synthetic relational data via probabilistic relational models. In particular, we propose a fully-fledged pipeline to go from relational database to probabilistic relational model, which can then be used to sample new synthetic relational data points from its underlying probability distribution. As part of our proposed pipeline, we introduce a learning algorithm to construct a probabilistic relational model from a given relational database.

There is a growing need for pluralistic alignment methods that can steer language models towards individual attributes and preferences. One such method, Self-Supervised Alignment with Mutual Information (SAMI), uses conditional mutual information to encourage the connection between behavioral preferences and model responses. We conduct two experiments exploring SAMI in multi-task settings. First, we compare SAMI to Direct Preference Optimization (DPO) on a multi-task benchmark (MT-Bench), using a stronger model to generate training data for a weaker one across diverse categories (humanities, STEM, extraction, coding, math, reasoning, and roleplay). Our results indicate that one iteration of SAMI has a 57% win rate against DPO, with significant variation in performance between task categories. Second, we examine SAMI's impact on mathematical accuracy (GSM-8K) relative to supervised fine-tuning (SFT). While SAMI increases zero-shot performance by 1.1%, SFT is more effective with a 3.2% boost. However, SAMI shows interesting scaling trends. When given 10 attempts, SAMI improves accuracy by 3.9%, while SFT achieves a 10.1% increase. Combining SAMI with SFT yields an additional improvement of 1.3% in multi-attempt settings, though single-attempt accuracy remains unchanged.

To effectively study complex causal systems, it is often useful to construct representations that simplify parts of the system by discarding irrelevant details while preserving key features. The Information Bottleneck (IB) method is a widely used approach in representation learning that compresses random variables while retaining information about a target variable. Traditional methods like IB are purely statistical and ignore underlying causal structures, making them ill-suited for causal tasks. We propose the Causal Information Bottleneck (CIB), a causal extension of the IB, which compresses a set of chosen variables while maintaining causal control over a target variable. This method produces representations which are causally interpretable, and which can be used when reasoning about interventions. We present experimental results demonstrating that the learned representations accurately capture causality as intended.

We propose a theory for matrix completion that goes beyond the low-rank structure commonly considered in the literature and applies to general matrices of low description complexity. Specifically, complexity of the sets of matrices encompassed by the theory is measured in terms of Hausdorff and upper Minkowski dimensions. Our goal is the characterization of the number of linear measurements, with an emphasis on rank-$1$ measurements, needed for the existence of an algorithm that yields reconstruction, either perfect, with probability 1, or with arbitrarily small probability of error, depending on the setup. Concretely, we show that matrices taken from a set $\mathcal{U}$ such that $\mathcal{U}-\mathcal{U}$ has Hausdorff dimension $s$ can be recovered from $k>s$ measurements, and random matrices supported on a set $\mathcal{U}$ of Hausdorff dimension $s$ can be recovered with probability 1 from $k>s$ measurements. What is more, we establish the existence of recovery mappings that are robust against additive perturbations or noise in the measurements. Concretely, we show that there are $\beta$-H\"older continuous mappings recovering matrices taken from a set of upper Minkowski dimension $s$ from $k>2s/(1-\beta)$ measurements and, with arbitrarily small probability of error, random matrices supported on a set of upper Minkowski dimension $s$ from $k>s/(1-\beta)$ measurements. The numerous concrete examples we consider include low-rank matrices, sparse matrices, QR decompositions with sparse R-components, and matrices of fractal nature.

Common practice in modern machine learning involves fitting a large number of parameters relative to the number of observations. These overparameterized models can exhibit surprising generalization behavior, e.g., ``double descent'' in the prediction error curve when plotted against the raw number of model parameters, or another simplistic notion of complexity. In this paper, we revisit model complexity from first principles, by first reinterpreting and then extending the classical statistical concept of (effective) degrees of freedom. Whereas the classical definition is connected to fixed-X prediction error (in which prediction error is defined by averaging over the same, nonrandom covariate points as those used during training), our extension of degrees of freedom is connected to random-X prediction error (in which prediction error is averaged over a new, random sample from the covariate distribution). The random-X setting more naturally embodies modern machine learning problems, where highly complex models, even those complex enough to interpolate the training data, can still lead to desirable generalization performance under appropriate conditions. We demonstrate the utility of our proposed complexity measures through a mix of conceptual arguments, theory, and experiments, and illustrate how they can be used to interpret and compare arbitrary prediction models.

This work uniquely identifies and characterizes four prevalent multimodal model architectural patterns in the contemporary multimodal landscape. Systematically categorizing models by architecture type facilitates monitoring of developments in the multimodal domain. Distinct from recent survey papers that present general information on multimodal architectures, this research conducts a comprehensive exploration of architectural details and identifies four specific architectural types. The types are distinguished by their respective methodologies for integrating multimodal inputs into the deep neural network model. The first two types (Type A and B) deeply fuses multimodal inputs within the internal layers of the model, whereas the following two types (Type C and D) facilitate early fusion at the input stage. Type-A employs standard cross-attention, whereas Type-B utilizes custom-designed layers for modality fusion within the internal layers. On the other hand, Type-C utilizes modality-specific encoders, while Type-D leverages tokenizers to process the modalities at the model's input stage. The identified architecture types aid the monitoring of any-to-any multimodal model development. Notably, Type-C and Type-D are currently favored in the construction of any-to-any multimodal models. Type-C, distinguished by its non-tokenizing multimodal model architecture, is emerging as a viable alternative to Type-D, which utilizes input-tokenizing techniques. To assist in model selection, this work highlights the advantages and disadvantages of each architecture type based on data and compute requirements, architecture complexity, scalability, simplification of adding modalities, training objectives, and any-to-any multimodal generation capability.

The success of AI models relies on the availability of large, diverse, and high-quality datasets, which can be challenging to obtain due to data scarcity, privacy concerns, and high costs. Synthetic data has emerged as a promising solution by generating artificial data that mimics real-world patterns. This paper provides an overview of synthetic data research, discussing its applications, challenges, and future directions. We present empirical evidence from prior art to demonstrate its effectiveness and highlight the importance of ensuring its factuality, fidelity, and unbiasedness. We emphasize the need for responsible use of synthetic data to build more powerful, inclusive, and trustworthy language models.