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In this work, we present a Linear Matrix Inequality (LMI) based method to synthesize an optimal H1 estimator for a large class of linear coupled partial differential equations (PDEs) utilizing only finite dimensional measurements. Our approach extends the newly developed framework for representing and analyzing distributed parameter systems using operators on the space of square integrable functions that are equipped with multipliers and kernels of semi-separable class. We show that by redefining the state, the PDEs can be represented using operators that embed the boundary conditions and input-output relations explicitly. The optimal estimator synthesis problem is formulated as a convex optimization subject to LMIs that require no approximation or discretization. A scalable algorithm is presented to synthesize the estimator. The algorithm is illustrated by suitable examples.
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This content will become publicly available on March 1, 2026
Linear Relational Decoding of Morphology in Language Models
A two-part affine approximation has been found to be a good approximation for trans- former computations over certain subject- object relations. Adapting the Bigger Analogy Test Set, we show that the linear transforma- tion W s, where s is a middle layer representa- tion of a subject token and W is derived from model derivatives, is also able to accurately re- produce final object states for many relations. This linear technique is able to achieve 90% faithfulness on morphological relations, and we show similar findings multi-lingually and across models. Our findings indicate that some conceptual relationships in language models, such as morphology, are readily interpretable from latent space, and are sparsely encoded by cross-layer linear transformations.
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- Award ID(s):
- 2349452
- PAR ID:
- 10655817
- Publisher / Repository:
- NAACL, aclanthology.org
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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