The collapsing function in Connectionist Temporal Classification (CTC) transforms alignment to the final output sequence.  It has two steps:

1) Remove any consecutive letters. For example, “dinner” is turned into “diner”

2) Remove blanks. For example, if we use “_” to denote blanks, “din_ner” is turned into “dinner”.

As a result, the collapsing function is a mapping from many to one. For example, “dinn_ner” and “diin_nne_r” both map to “dinner”.

University of Notre Dame

The alignment is a sequence of vectors where each vector represents a letter. It is the middle output of the Connectionist Temporal Classification (CTC) algorithm, and it has the same length as the input of CTC algorithm. 

Alignment in Connectionist Temporal Classification

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

Speech and Language Processing (3rd ed. draft) 

Collapsing Function

In Connectionist Temporal Classification (CTC), we assume that the output vectors in a single sequence are all independent. Mathematically, let the input sequence be $X=\{x_1, x_2,..,x_n\}$ where $x_i$ are vectors, and let the output alignment be $A=\{a_1, a_2, …, a_n\}$ where $a_i$ are vectors, then $P_{CTC}(A|X)=\prod_{i=1}^{n}p(a_i|X)$.

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