Most real world data sets consist of data vectors whose individual components are not statistically independent. In other words, knowing the value of an element will provide information about the value of elements in the data vector. When this occurs, it can be desirable to create a factorial code of the data, i.e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector (loss-free coding), but the code components are statistically independent.
Later supervised learning usually works much better when the raw input data is first translated into such a factorial code. For example, suppose the final goal is to classify images with highly redundant pixels. A naive Bayes classifier will assume the pixels are statistically independent random variables and therefore fail to produce good results. If the data are first encoded in a factorial way, however, then the naive Bayes classifier will achieve its optimal performance (compare Schmidhuber et al. 1996).
To create factorial codes, Horace Barlow and co-workers suggested to minimize the sum of the bit entropies of the code components of binary codes (1989). Jürgen Schmidhuber (1992) re-formulated the problem in terms of predictors and binary feature detectors, each receiving the raw data as an input. For each detector there is a predictor that sees the other detectors and learns to predict the output of its own detector in response to the various input vectors or images. But each detector uses a machine learning algorithm to become as unpredictable as possible. The global optimum of this objective function corresponds to a factorial code represented in a distributed fashion across the outputs of the feature detectors.
Painsky, Rosset and Feder (2016, 2017) further studied this problem in the context of independent component analysis over finite alphabet sizes. Through a series of theorems they show that the factorial coding problem can be accurately solved with a branch and bound search tree algorithm, or tightly approximated with a series of linear problems. In addition, they introduce a simple transformation (namely, order permutation) which provides a greedy yet very effective approximation of the optimal solution. Practically, they show that with a careful implementation, the favorable properties of the order permutation may be achieved in an asymptotically optimal computational complexity. Importantly, they provide theoretical guarantees, showing that while not every random vector can be efficiently decomposed into independent components, the majority of vectors do decompose very well (that is, with a small constant cost), as the dimension increases. In addition, they demonstrate the use of factorial codes to data compression in multiple setups (2017).
a factorialcode of the data, i.e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector...
In mathematics, the factorial of a non-negative integer n {\displaystyle n} , denoted by n ! {\displaystyle n!} , is the product of all positive integers...
the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base...
In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels"...
fractional factorial designs are experimental designs consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design...
the for statement. counter, factorial := 5, 1 for counter > 1 { counter, factorial = counter-1, factorial*counter } The code for the loop is the same for...
factorial: function factorial(n) if n == 0 return 1 else return n * factorial(n - 1) end end Indeed, n * factorial(n - 1) wraps the call to factorial...
analysis Factor regression model Factor graph FactorialcodeFactorial experiment Factorial moment Factorial moment generating function Failure rate Fair...
Function (fix) factorial = fix fac where fac f x | x < 2 = 1 | otherwise = x * f (x - 1) As the Integer type has arbitrary-precision, this code will compute...
code Linear-feedback shift register De Bruijn sequence Steinhaus–Johnson–Trotter algorithm – an algorithm that generates Gray codes for the factorial...
a factorialcode of the data, i.e., a new vector-valued representation of each data vector such that it gets uniquely encoded by the resulting code vector...
second "factorial" message, yielding 3628800. 3628800 then receives "between:and:", answering false. Because the meaning of binary messages is not coded into...
memoized version of factorial using the above strategy, rather than calling factorial directly, code invokes memoized-call(factorial(n)). Each such call...
accepted by the emerging field of psychology which developed strong (full factorial) experimental methods to which randomization and blinding were soon added...
used in code as any other fudget. factorialF = stdoutF >==< mapF (show . factorial . read) >==< stdinF factorial :: Integer -> Integer factorial n = product...
the factorial function for non-negative integers, shown in Haskell: factorial :: Integer -> Integer factorial 0 = 1 factorial n = n * factorial (n-1)...
acceptable mathematically. But different factorial theories proved to differ as much in terms of the orientations of factorial axes for a given solution as in...
section of code. For example: The result of a factorial is always an integer and greater than or equal to 1. So a program that calculates the factorial of an...
used for abstraction. The factorial function can be expressed as follows: fun factorial n = if n = 0 then 1 else n * factorial (n - 1) An SML compiler must...
\cdots \times (n-k+1)}{k\times (k-1)\times \cdots \times 1}},} which using factorial notation can be compactly expressed as ( n k ) = n ! k ! ( n − k ) ! ...
algorithmic way of constructing such a function - searching for one is a factorial function of the number of keys to be mapped versus the number of table...
three-level factorial experiment. After the designed experiment is performed, linear regression is used, sometimes iteratively, to obtain results. Coded variables...
Global Information