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The stochastic actor oriented model (SAOM) is a method for modelling social interactions and social behaviour over time. It can be used to model drivers of dynamic interactions using both exogenous covariates and endogenous network configurations, but also the co-evolution of behaviour and social interactions. In its standard implementations, it assumes that all individual have the same interaction evaluation function. This lack of heterogeneity is one of its limitations. The aim of this paper is to extend the inference framework for the SAOM to include random effects, so that the heterogeneity of individuals can be modeled more accurately. We decompose the linear evaluation function that models the probability of forming or removing a tie from the network, in a homogeneous fixed part and a random, individual-specific part. We extend the Robbins-Monro algorithm to the estimation of the variance of the random parameters. Our method is applicable for the general random effect formulations. We illustrate the method with a random out-degree model and show the parameter estimation of the random components, significance tests and model evaluation. We apply the method to the Kapferer's Tailor shop study. It is shown that a random out-degree constitutes a serious alternative to including transitivity and higher-order dependency effects.

Few neural architectures lend themselves to provable learning with gradient based methods. One popular model is the single-index model, in which labels are produced by composing an unknown linear projection with a possibly unknown scalar link function. Learning this model with SGD is relatively well-understood, whereby the so-called information exponent of the link function governs a polynomial sample complexity rate. However, extending this analysis to deeper or more complicated architectures remains challenging. In this work, we consider single index learning in the setting of symmetric neural networks. Under analytic assumptions on the activation and maximum degree assumptions on the link function, we prove that gradient flow recovers the hidden planted direction, represented as a finitely supported vector in the feature space of power sum polynomials. We characterize a notion of information exponent adapted to our setting that controls the efficiency of learning.

We introduce Adaptive Functional Principal Component Analysis, a novel method to capture directions of variation in functional data that exhibit sharp changes in smoothness. We first propose a new adaptive scatterplot smoothing technique that is fast and scalable, and then integrate this technique into a probabilistic FPCA framework to adaptively smooth functional principal components. Our simulation results show that our approach is better able to model functional data with sharp changes in smoothness compared to standard approaches. We are motivated by the need to identify coordinated patterns of brain activity across multiple neurons during reaching movements prompted by an auditory cue, which enables understanding of the dynamics in the brain during dexterous movement. Our proposed method captures the underlying biological mechanisms that arise in data obtained from a mouse experiment focused on voluntary reaching movements, offering more interpretable activation patterns that reflect sharp changes in neural activity following the cue. We develop accompanying publicly available software for our proposed methodology, along with implementations to reproduce our results.

Given the success of Large Language Models (LLMs), there has been considerable interest in studying the properties of model activations. The literature overwhelmingly agrees that LLM representations are dominated by a few ``outlier dimensions'' with exceedingly high variance and magnitude. Several studies in Natural Language Processing (NLP) have sought to mitigate the impact of such outlier dimensions and force LLMs to be isotropic (i.e., have uniform variance across all dimensions in embedding space). Isotropy is thought to be a desirable property for LLMs that improves model performance and more closely aligns textual representations with human intuition. However, many of the claims regarding isotropy in NLP have been based on the average cosine similarity of embeddings, which has recently been shown to be a flawed measure of isotropy. In this paper, we propose I-STAR: IsoScore*-based STable Anisotropic Regularization, a novel regularization method that can be used to increase or decrease levels of isotropy in embedding space during training. I-STAR uses IsoScore*, the first accurate measure of isotropy that is both differentiable and stable on mini-batch computations. In contrast to several previous works, we find that decreasing isotropy in contextualized embeddings improves performance on the majority of tasks and models considered in this paper.

We study the complexity of learning quantum states in various models with respect to the stabilizer formalism and obtain the following results: - We prove that $\Omega(n)$ $T$-gates are necessary for any Clifford+$T$ circuit to prepare computationally pseudorandom quantum states, an exponential improvement over the previously known bound. This bound is asymptotically tight if linear-time quantum-secure pseudorandom functions exist. - Given an $n$-qubit pure quantum state $|\psi\rangle$ that has fidelity at least $\tau$ with some stabilizer state, we give an algorithm that outputs a succinct description of a stabilizer state that witnesses fidelity at least $\tau - \varepsilon$. The algorithm uses $O(n/(\varepsilon^2\tau^4))$ samples and $\exp\left(O(n/\tau^4)\right) / \varepsilon^2$ time. In the regime of $\tau$ constant, this algorithm estimates stabilizer fidelity substantially faster than the na\"ive $\exp(O(n^2))$-time brute-force algorithm over all stabilizer states. - In the special case of $\tau > \cos^2(\pi/8)$, we show that a modification of the above algorithm runs in polynomial time. - We improve the soundness analysis of the stabilizer state property testing algorithm due to Gross, Nezami, and Walter [Comms. Math. Phys. 385 (2021)]. As an application, we exhibit a tolerant property testing algorithm for stabilizer states. The underlying algorithmic primitive in all of our results is Bell difference sampling. To prove our results, we establish and/or strengthen connections between Bell difference sampling, symplectic Fourier analysis, and graph theory.

Activation functions are essential to introduce nonlinearity into neural networks, with the Rectified Linear Unit (ReLU) often favored for its simplicity and effectiveness. Motivated by the structural similarity between a shallow Feedforward Neural Network (FNN) and a single iteration of the Projected Gradient Descent (PGD) algorithm, a standard approach for solving constrained optimization problems, we consider ReLU as a projection from R onto the nonnegative half-line R+. Building on this interpretation, we extend ReLU by substituting it with a generalized projection operator onto a convex cone, such as the Second-Order Cone (SOC) projection, thereby naturally extending it to a Multivariate Projection Unit (MPU), an activation function with multiple inputs and multiple outputs. We further provide a mathematical proof establishing that FNNs activated by SOC projections outperform those utilizing ReLU in terms of expressive power. Experimental evaluations on widely-adopted architectures further corroborate MPU's effectiveness against a broader range of existing activation functions.

Diffusion models are powerful generative models that map noise to data using stochastic processes. However, for many applications such as image editing, the model input comes from a distribution that is not random noise. As such, diffusion models must rely on cumbersome methods like guidance or projected sampling to incorporate this information in the generative process. In our work, we propose Denoising Diffusion Bridge Models (DDBMs), a natural alternative to this paradigm based on diffusion bridges, a family of processes that interpolate between two paired distributions given as endpoints. Our method learns the score of the diffusion bridge from data and maps from one endpoint distribution to the other by solving a (stochastic) differential equation based on the learned score. Our method naturally unifies several classes of generative models, such as score-based diffusion models and OT-Flow-Matching, allowing us to adapt existing design and architectural choices to our more general problem. Empirically, we apply DDBMs to challenging image datasets in both pixel and latent space. On standard image translation problems, DDBMs achieve significant improvement over baseline methods, and, when we reduce the problem to image generation by setting the source distribution to random noise, DDBMs achieve comparable FID scores to state-of-the-art methods despite being built for a more general task.

Existing Collaborative Filtering (CF) methods are mostly designed based on the idea of matching, i.e., by learning user and item embeddings from data using shallow or deep models, they try to capture the associative relevance patterns in data, so that a user embedding can be matched with relevant item embeddings using designed or learned similarity functions. However, as a cognition rather than a perception intelligent task, recommendation requires not only the ability of pattern recognition and matching from data, but also the ability of cognitive reasoning in data. In this paper, we propose to advance Collaborative Filtering (CF) to Collaborative Reasoning (CR), which means that each user knows part of the reasoning space, and they collaborate for reasoning in the space to estimate preferences for each other. Technically, we propose a Neural Collaborative Reasoning (NCR) framework to bridge learning and reasoning. Specifically, we integrate the power of representation learning and logical reasoning, where representations capture similarity patterns in data from perceptual perspectives, and logic facilitates cognitive reasoning for informed decision making. An important challenge, however, is to bridge differentiable neural networks and symbolic reasoning in a shared architecture for optimization and inference. To solve the problem, we propose a modularized reasoning architecture, which learns logical operations such as AND ($\wedge$), OR ($\vee$) and NOT ($\neg$) as neural modules for implication reasoning ($\rightarrow$). In this way, logical expressions can be equivalently organized as neural networks, so that logical reasoning and prediction can be conducted in a continuous space. Experiments on real-world datasets verified the advantages of our framework compared with both shallow, deep and reasoning models.

Attention networks in multimodal learning provide an efficient way to utilize given visual information selectively. However, the computational cost to learn attention distributions for every pair of multimodal input channels is prohibitively expensive. To solve this problem, co-attention builds two separate attention distributions for each modality neglecting the interaction between multimodal inputs. In this paper, we propose bilinear attention networks (BAN) that find bilinear attention distributions to utilize given vision-language information seamlessly. BAN considers bilinear interactions among two groups of input channels, while low-rank bilinear pooling extracts the joint representations for each pair of channels. Furthermore, we propose a variant of multimodal residual networks to exploit eight-attention maps of the BAN efficiently. We quantitatively and qualitatively evaluate our model on visual question answering (VQA 2.0) and Flickr30k Entities datasets, showing that BAN significantly outperforms previous methods and achieves new state-of-the-arts on both datasets.

We investigate a lattice-structured LSTM model for Chinese NER, which encodes a sequence of input characters as well as all potential words that match a lexicon. Compared with character-based methods, our model explicitly leverages word and word sequence information. Compared with word-based methods, lattice LSTM does not suffer from segmentation errors. Gated recurrent cells allow our model to choose the most relevant characters and words from a sentence for better NER results. Experiments on various datasets show that lattice LSTM outperforms both word-based and character-based LSTM baselines, achieving the best results.