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SUMMARY:Kirk Lecture: Bridging the Divide: from Matrix to Tensor algebra f
 or Optimal Approximation and Compression - Misha Kilmer (Tufts University)
DTSTART:20230602T150000Z
DTEND:20230602T160000Z
UID:TALK201514@talks.cam.ac.uk
DESCRIPTION:Tensors\, also known as multiway arrays\, have become ubiquito
 us as representations for operators or as convenient schemes for storing d
 ata. Yet\, when it comes to compressing these objects or analyzing the dat
 a stored in them\, the tendency is to ``flatten&rdquo\; or ``matricize&rdq
 uo\; the data and employ traditional linear algebraic tools\, ignoring hig
 her dimensional correlations/structure that could have been exploited. Imp
 ediments to the development of equivalent tensor-based approaches stem fro
 m the fact that familiar concepts\, such as rank and orthogonal decomposit
 ion\, have no straightforward analogues and/or lead to intractable computa
 tional problems for tensors of order three and higher. In this talk\, we w
 ill review some of the common tensor decompositions and discuss their theo
 retical and practical limitations. We then discuss a family of tensor alge
 bras based on a new definition of tensor-tensor products. Unlike other ten
 sor approaches\, the framework we derive based around this tensor-tensor p
 roduct allows us to generalize in a very elegant way all classical algorit
 hms from linear algebra. Furthermore\, under our framework\, tensors can b
 e decomposed in a natural (e.g. &lsquo\;matrix-mimetic&rsquo\;) way with p
 rovable approximation properties and with provable benefits over tradition
 al matrix approximation. In addition to several examples from recent liter
 ature illustrating the advantages of our tensor-tensor product framework i
 n practice\, we highlight interesting open questions and directions for fu
 ture research.
LOCATION:Seminar Room 1\, Newton Institute
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