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Mathematics
Linear and Abstract Algebra
What is the difference between a complete basis and an overcomplete dictionary?
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[QUOTE="fog37, post: 6629030, member: 503639"] Thank you pbuk! I really like your postcode example. Just checking my understanding, hopefully useful for other beginners like me who don't fully appreciate the power of frame and dictionaries... [LIST] [*]From what I understand, a frame is like the generalized version of a basis. A basis, a particular type of frame, is a complete set of independent vectors. Not all frames are bases but all bases are frames. A frame is generally a complete set of dependent vectors (essentially a non-orthogonal basis). [*]"Complete" means that the basis or frame has as many basis/frame vectors as the dimension of the space a vector lives in: if our vector ##X## lives in ##R3##, then any complete frame or basis would have 3 building block vectors. [*]Using an analogy, frames are like the letters of an alphabet (and there are many alphabets). [*]When we use complete but non-orthogonal bases, a hypothetical vector ##X## does still has a unique representation (linear superposition of the non orthogonal basis vectors) as it happens when we use bases but the coefficients cannot be easily calculated via dot products. [*]My understanding is that "dictionary" and "frame" are synonyms. [*]An [U]overcomplete dictionary[/U] seems to be a frame with a number of dependent vectors that is larger than the dimensionality of the vectors we are interested in representing. Using an overcomplete dictionary, a vector ##X## does NOT have a unique representation but multiple valid representations (redundancy). Also, we don't have to use all the atoms in the dictionary to represent the vector (something we need to do when we use a basis). [/LIST] The interesting part (at least for me), is that certain types of vectors can have a sparser representation in an overcomplete dictionary than in a complete basis or frame. For example, let's consider 2 different alphabets. Alphabet 1 has 5 letters and is an overcomplete dictionary with the following atoms [B]$,@,%,&,*[/B] Alphabet 2 has letters and is a complete basis with the atoms/letters [B]O,U,W.[/B] We could write the same word using either alphabet but we would, somehow, get both a redundant (i.e. multiple linear expansions are possible) and sparser representation using alphabet 1 even if alphabet 2 is shorter. That is cool. Thank you! [/QUOTE]
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Linear and Abstract Algebra
What is the difference between a complete basis and an overcomplete dictionary?
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