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An embedding, or a smooth embedding, is defined to be an immersion that is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image). [ 4 ] In other words, the domain of an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold .
A smooth embedding is an injective immersion f : M → N that is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is precisely a local embedding – that is, for any point x ∈ M there is a neighbourhood, U ⊆ M, of x such that f : U → N is an embedding, and conversely a local embedding is an ...
A latent space, also known as a latent feature space or embedding space, is an embedding of a set of items within a manifold in which items resembling each other are positioned closer to one another. Position within the latent space can be viewed as being defined by a set of latent variables that emerge from the resemblances from the objects.
An embedded graph uniquely defines cyclic orders of edges incident to the same vertex. The set of all these cyclic orders is called a rotation system.Embeddings with the same rotation system are considered to be equivalent and the corresponding equivalence class of embeddings is called combinatorial embedding (as opposed to the term topological embedding, which refers to the previous ...
In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 {\displaystyle \mathbb {R} ^{3}} upon itself (known as an ambient isotopy ); these transformations correspond to manipulations of a knotted ...
In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space. Several of the Sobolev embedding theorems are continuous embedding theorems.
The embedding of X into Y is a compact operator: any bounded set in X is totally bounded in Y, i.e. every sequence in such a bounded set has a subsequence that is Cauchy in the norm ||•|| Y. If Y is a Banach space, an equivalent definition is that the embedding operator (the identity) i : X → Y is a compact operator.
In natural language processing, a word embedding is a representation of a word. The embedding is used in text analysis . Typically, the representation is a real-valued vector that encodes the meaning of the word in such a way that the words that are closer in the vector space are expected to be similar in meaning. [ 1 ]