In this work, we give a poly(d,k) time and sample algorithm for efficiently learning the parameters (i.e., the means and the mixture weights) of a mixture of k spherical distributions in d dimensions. Unlike all previous methods, our techniques apply to heavy-tailed distributions and include examples that do not even have finite covariances. Our method succeeds whenever the component distributions have a characteristic function with sufficiently heavy tails. Examples of such distributions include the Laplace distribution and uniform over [−1, 1] but crucially exclude Gaussians.

In this work, we consider the problem of learning one hidden layer ReLU neural networks with inputs from $\mathbb{R}^d$. It is well known due to (Klivans and Sherstov, 2009) that without further assumptions on the distribution $\mathcal{D}$, e.g., when $\mathcal{D}$ can be supported over the Boolean hypercube, learning even one-hidden layer neural networks is impossible (or “hard”) for polynomial-time estimators under standard cryptographic assumptions. Given the success of neural networks in practice, a long line of recent work has attempted to study instead the canonical continuous input distribution case where $\mathcal{D}$ is the isotropic Gaussian, i.e., $\mathcal{D}=\mathcal{N}\left(0,I_d\right)$ which is also the setting that we follow in this work. Yet, despite a long line of research, it remains open whether there is a polynomial-time algorithm for learning one hidden layer neural networks when $\mathcal{D} = \mathcal{N}\left(0,I_d\right)$. It is known that a single neuron, i.e., zero hidden layer neural network, can be learned in polynomial time (Zarifis et al., 2024), while neural networks with more than two hidden layers are hard to learn (Chen et al., 2022). Nevertheless, the case of one hidden layer neural networks is not well understood. In this paper we close this gap in the literature by answering the question of efficient learnability of neural networks with one hidden layer. We establish that under the CLWE assumption from cryptography (Bruna et al., 2021), learning the class of one hidden layer neural network with polynomial size under standard Gaussian inputs and polynomially small Gaussian noise is indeed computationally hard. Importantly, solving CLWE in polynomial time implies a polynomial-time quantum algorithm that solves the worst-case gap shortest vector problem (GapSVP) within polynomial factors, a widely believed hard task in cryptography and algorithmic theory of lattices (Micciancio and Regev, 2009). En route, we prove the hardness of learning Lipschitz periodic functions under standard Gaussian inputs and polynomially small Gaussian noise. This improves the previous result from (Song et al., 2021), which proved the hardness for polynomially small adversarial noise. We also utilize the more general reductions between CLWE and classical LWE due to (Gupte et al., 2022). In particular, we show that if we assume the hardness of GapSVP with subexponential approximation factors $2^{O(d^{\delta })}$ for $\delta \in (0, 1)$, we can show the hardness of learning one hidden layer neural networks with polynomial size under Gaussian noise with $2^{-d^{\eta}}$ variance, where $\eta = \frac{\delta}{1+\delta}\in (0, 1/2)$. The current state-of-the-art algorithm for GapSVP is the celebrated Lenstra-Lenstra-Lovász (LLL) lattice basis reduction algorithm (Lenstra et al., 1982) which has approximation factor $2^{\Theta(d)}$. Hence, our results show that any polynomial time learning algorithm for one hidden layer neural networks for any variance of noise $\sigma^2 \ge 2^{-o(\sqrt{d})}$ would imply a major algorithmic breakthrough in the theory of lattices.

Prescriptions play an essential role in the process of Traditional Chinese Medicine (TCM) diagnosis and treatment. Prescription generation is to generate a set of herbs to treat the symptoms of a patient by analyzing the relationship between symptoms and herbs. Although there have been a couple of studies to generate prescriptions, they have ignored the implicit relationship between the different symptoms of the patients. In addition, abundant semantic information and interpretability of the knowledge graph can help to portray the complicated relationships between the various modules of TCM. Therefore, this paper proposes a Knowledge-Aware neural Group representation learning model for Attentive Prescription Generation of Traditional Chinese Medicine (KGAPG), which regards the prescription generation task as a group recommendation problem. More specifically, multiple symptoms of a patient are considered as a symptom group and the complicated semantic information between symptoms and herbs can be captured by the knowledge graph. The syndrome information of multiple symptoms which is summarized by a group aggregation method based on the attention mechanism is applied to simulate the actual process of TCM diagnosis and treatment. The experiment results demonstrate that KGAPG is effective on a TCM prescription benchmark dataset, and its evaluation indicators of Precision, Recall and NDCG exceed other state-of-the-art methods.

Abstract The rapid growth of event‐based social networks (EBSNs) has generated great demand for personalized event recommendation. Indeed, new events are published every day in EBSNs. In this paper, we focus on the problem of recommending new events (i.e., the newly published events that have not yet received any user response) to registered users in EBSNs. Notice that collaborative filtering is ineffective for recommending new events due to the short lifetime of events and the lack of historical information, that is, the cold‐start problem. A straightforward approach to address this problem is to adopt content‐based recommendation techniques, for example, deriving user interests through the content information of the events they attended. Nevertheless, we observe that organizer influence and geographical preference also play important roles in a user's decision to participate in an event. Motivated by this observation, we combine user interest, organizer influence, and geographical preference to present a unified model for new event recommendation. In particular, we utilize a topic model to derive user interests while adopting matrix factorization to infer organizer influences on users from the interactions between users and organizers. Moreover, we model users' geographical preferences via their location histories of attended events. We conduct a performance evaluation using real‐world data collected from DoubanEvent. The experimental results demonstrate the effectiveness of our proposed recommendation model.