Riemannian Optimization for Hadamard Products of Low-Rank Matrices

The elementwise Hadamard product of two low-rank matrices provides a parameter-efficient model for data with multiplicative structure, but its modeling is challenging due to the presence of additional symmetries under coupled row/column scalings between the two factors. In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold. We propose a novel block-diagonal Riemannian metric derived from the pullback of the Frobenius inner product. The metric is shown to be invariant under the full symmetry group. We develop a Riemannian gradient descent algorithm that uses a tuning-free Gauss--Newton step size and scales linearly in the number of observed entries per iteration. Experiments on real and synthetic datasets illustrate the efficacy of our proposed Riemannian approach.

SSP: Self-Supervised Prompting for Cross-Lingual Transfer to Low-Resource Languages using Large Language Models

Recently, very large language models (LLMs) have shown exceptional performance on several English NLP tasks with just in-context learning (ICL), but their utility in other languages is still underexplored. We investigate their effectiveness for NLP tasks in low-resource languages (LRLs), especially in the setting of zero-labelled cross-lingual transfer (0-CLT), where no labelled training data for the target language is available -- however training data from one or more related medium-resource languages (MRLs) is utilized, alongside the available unlabeled test data for a target language. We introduce Self-Supervised Prompting (SSP), a novel ICL approach tailored for the 0-CLT setting. SSP is based on the key observation that LLMs output more accurate labels if in-context exemplars are from the target language (even if their labels are slightly noisy). To operationalize this, since target language training data is not available in 0-CLT, SSP operates in two stages. In Stage I, using source MRL training data, target language's test data is noisily labeled. In Stage II, these noisy test data points are used as exemplars in ICL for further improved labelling. Additionally, our implementation of SSP uses a novel Integer Linear Programming (ILP)-based exemplar selection that balances similarity, prediction confidence (when available) and label coverage. Experiments on three tasks and eleven LRLs (from three regions) demonstrate that SSP strongly outperforms existing SOTA fine-tuned and prompting-based baselines in 0-CLT setup.