Editing Independent component analysis (section)

== Introduction ==
[[File:A-Local-Learning-Rule-for-Independent-Component-Analysis-srep28073-s3.ogv|300px|thumb|right|ICA on four randomly mixed videos.<ref>{{cite journal|last1=Isomura|first1=Takuya|last2=Toyoizumi|first2=Taro|title=A local learning rule for independent component analysis|journal=Scientific Reports|volume=6|pages=28073|date=2016|doi=10.1038/srep28073|pmid=27323661|pmc=4914970|bibcode=2016NatSR...628073I}}</ref> Top row: The original source videos. Middle row: Four random mixtures used as input to the algorithm. Bottom row: The reconstructed videos.]]
Independent component analysis attempts to decompose a multivariate signal into independent non-Gaussian signals. As an example, sound is usually a signal that is composed of the numerical addition, at each time t, of signals from several sources. The question then is whether it is possible to separate these contributing sources from the observed total signal. When the statistical independence assumption is correct, blind ICA separation of a mixed signal gives very good results.<ref name="ComoJ2010">Comon, P.; Jutten C., (2010): Handbook of Blind Source Separation, Independent Component Analysis and Applications. Academic Press, Oxford UK. {{ISBN|978-0-12-374726-6}}</ref> It is also used for signals that are not supposed to be generated by mixing for analysis purposes.

A simple application of ICA is the "[[cocktail party problem]]", where the underlying speech signals are separated from a sample data consisting of people talking simultaneously in a room. Usually the problem is simplified by assuming no time delays or echoes. Note that a filtered and delayed signal is a copy of a dependent component, and thus the statistical independence assumption is not violated.

Mixing weights for constructing the ''<math display="inline">M</math>'' observed signals from the <math display="inline">N</math> components can be placed in an <math display="inline">M \times N</math> matrix. An important thing to consider is that if <math display="inline">N</math> sources are present, at least <math display="inline">N</math> observations (e.g. microphones if the observed signal is audio) are needed to recover the original signals. When there are an equal number of observations and source signals, the mixing matrix is square (''<math display="inline">M = N</math>''). Other cases of underdetermined (''<math display="inline">M < N</math>'') and overdetermined (''<math display="inline">M > N</math>'') have been investigated.

The success of ICA separation of mixed signals relies on two assumptions and three effects of mixing source signals. Two assumptions:
#The source signals are independent of each other.
#The values in each source signal have non-Gaussian distributions.
Three effects of mixing source signals:
#Independence: As per assumption 1, the source signals are independent; however, their signal mixtures are not. This is because the signal mixtures share the same source signals.
#Normality: According to the [[Central Limit Theorem]], the distribution of a sum of independent random variables with finite variance tends towards a Gaussian distribution.<br />Loosely speaking, a sum of two independent random variables usually has a distribution that is closer to Gaussian than any of the two original variables. Here we consider the value of each signal as the random variable.
#Complexity: The temporal complexity of any signal mixture is greater than that of its simplest constituent source signal.
Those principles contribute to the basic establishment of ICA. If the signals extracted from a set of mixtures are independent and have non-Gaussian distributions or have low complexity, then they must be source signals.<ref name="Stone 2004">{{cite book|last=Stone|first=James V.|title=Independent component analysis : a tutorial introduction|year=2004|publisher=MIT Press|location=Cambridge, Massachusetts |isbn=978-0-262-69315-8}}</ref><ref>{{cite book|last1= Hyvärinen |first1=Aapo |last2=Karhunen |first2=Juha |last3=Oja |first3=Erkki |title=Independent component analysis|year=2001|publisher=John Wiley & Sons|location=New York|isbn=978-0-471-22131-9|edition=1st}}</ref>

Another common example is image [[steganography]], where ICA is used to embed one image within another. For instance, two grayscale images can be linearly combined to create mixed images in which the hidden content is visually imperceptible. ICA can then be used to recover the original source images from the mixtures. This technique underlies digital watermarking, which allows the embedding of ownership information into images, as well as more covert applications such as undetected information transmission. The method has even been linked to real-world cyberespionage cases. In such applications, ICA serves to unmix the data based on statistical independence, making it possible to extract hidden components that are not apparent in the observed data.

Steganographic techniques, including those potentially involving ICA-based analysis, have been used in real-world cyberespionage cases. In 2010, the FBI uncovered a Russian spy network known as the "Illegals Program" (Operation Ghost Stories), where agents used custom-built steganography tools to conceal encrypted text messages within image files shared online.<ref>{{cite web |title=Operation Ghost Stories: Inside the Russian Spy Case |url=https://www.fbi.gov/news/stories/operation-ghost-stories-inside-the-russian-spy-case |website=FBI.gov |publisher=Federal Bureau of Investigation |date=28 June 2010}}</ref>

In another case, a former General Electric engineer, Xiaoqing Zheng, was convicted in 2022 for economic espionage. Zheng used steganography to exfiltrate sensitive turbine technology by embedding proprietary data within image files for transfer to entities in China.<ref>{{cite web |title=Former GE Power Engineer Sentenced for Conspiracy to Commit Economic Espionage |url=https://www.justice.gov/archives/opa/pr/former-ge-power-engineer-sentenced-conspiracy-commit-economic-espionage |website=Justice.gov |publisher=U.S. Department of Justice |date=3 January 2022}}</ref>