Cover-coding is a technique for obscuring the data that is transmitted over an insecure link, to reduce the risks of snooping. An example of cover-coding would be for the sender to perform a bitwise XOR (exclusive OR) of the original data with a password or random number which is known to both sender and receiver. The resulting cover-coded data is then transmitted from sender to the receiver, who uncovers the original data by performing a further bitwise XOR (exclusive OR) operation on the received data using the same password or random number. ISO 18000-6C (EPC Class 1 Generation 2) RFID tags protect some operations with a cover code. The reader requests a random number from the tag, and the tag responds with a new random number. The reader then encrypts future communications with this number, using bitwise XOR, to the data it sends. Cover coding is secure if the tag signal can't be intercepted and the random number is not re-used. Compared to the loud transmissions from the reader, tag backscatter is much weaker and difficult -- but not impossible -- to intercept.
Sequence labeling
In machine learning, sequence labeling is a type of pattern recognition task that involves the algorithmic assignment of a categorical label to each member of a sequence of observed values. A common example of a sequence labeling task is part of speech tagging, which seeks to assign a part of speech to each word in an input sentence or document. Sequence labeling can be treated as a set of independent classification tasks, one per member of the sequence. However, accuracy is generally improved by making the optimal label for a given element dependent on the choices of nearby elements, using special algorithms to choose the globally best set of labels for the entire sequence at once. As an example of why finding the globally best label sequence might produce better results than labeling one item at a time, consider the part-of-speech tagging task just described. Frequently, many words are members of multiple parts of speech, and the correct label of such a word can often be deduced from the correct label of the word to the immediate left or right. For example, the word "sets" can be either a noun or verb. In a phrase like "he sets the books down", the word "he" is unambiguously a pronoun, and "the" unambiguously a determiner, and using either of these labels, "sets" can be deduced to be a verb, since nouns very rarely follow pronouns and are less likely to precede determiners than verbs are. But in other cases, only one of the adjacent words is similarly helpful. In "he sets and then knocks over the table", only the word "he" to the left is helpful (cf. "...picks up the sets and then knocks over..."). Conversely, in "... and also sets the table" only the word "the" to the right is helpful (cf. "... and also sets of books were ..."). An algorithm that proceeds from left to right, labeling one word at a time, can only use the tags of left-adjacent words and might fail in the second example above; vice versa for an algorithm that proceeds from right to left. Most sequence labeling algorithms are probabilistic in nature, relying on statistical inference to find the best sequence. The most common statistical models in use for sequence labeling make a Markov assumption, i.e. that the choice of label for a particular word is directly dependent only on the immediately adjacent labels; hence the set of labels forms a Markov chain. This leads naturally to the hidden Markov model (HMM), one of the most common statistical models used for sequence labeling. Other common models in use are the maximum entropy Markov model and conditional random field.
Sample complexity
The sample complexity of a machine learning algorithm represents the number of training-samples that it needs in order to successfully learn a target function. More precisely, the sample complexity is the number of training-samples that we need to supply to the algorithm, so that the function returned by the algorithm is within an arbitrarily small error of the best possible function, with probability arbitrarily close to 1. There are two variants of sample complexity: The weak variant fixes a particular input-output distribution; The strong variant takes the worst-case sample complexity over all input-output distributions. The No free lunch theorem, discussed below, proves that, in general, the strong sample complexity is infinite, i.e. that there is no algorithm that can learn the globally-optimal target function using a finite number of training samples. However, if we are only interested in a particular class of target functions (e.g., only linear functions) then the sample complexity is finite, and it depends linearly on the VC dimension on the class of target functions. == Definition == Let X {\displaystyle X} be a space which we call the input space, and Y {\displaystyle Y} be a space which we call the output space, and let Z {\displaystyle Z} denote the product X × Y {\displaystyle X\times Y} . For example, in the setting of binary classification, X {\displaystyle X} is typically a finite-dimensional vector space and Y {\displaystyle Y} is the set { − 1 , 1 } {\displaystyle \{-1,1\}} . Fix a hypothesis space H {\displaystyle {\mathcal {H}}} of functions h : X → Y {\displaystyle h\colon X\to Y} . A learning algorithm over H {\displaystyle {\mathcal {H}}} is a computable map from Z {\displaystyle Z} to H {\displaystyle {\mathcal {H}}} . In other words, it is an algorithm that takes as input a finite sequence of training samples and outputs a function from X {\displaystyle X} to Y {\displaystyle Y} . Typical learning algorithms include empirical risk minimization, without or with Tikhonov regularization. Fix a loss function L : Y × Y → R ≥ 0 {\displaystyle {\mathcal {L}}\colon Y\times Y\to \mathbb {R} _{\geq 0}} , for example, the square loss L ( y , y ′ ) = ( y − y ′ ) 2 {\displaystyle {\mathcal {L}}(y,y')=(y-y')^{2}} , where h ( x ) = y ′ {\displaystyle h(x)=y'} . For a given distribution ρ {\displaystyle \rho } on X × Y {\displaystyle X\times Y} , the expected risk of a hypothesis (a function) h ∈ H {\displaystyle h\in {\mathcal {H}}} is E ( h ) := E ρ [ L ( h ( x ) , y ) ] = ∫ X × Y L ( h ( x ) , y ) d ρ ( x , y ) {\displaystyle {\mathcal {E}}(h):=\mathbb {E} _{\rho }[{\mathcal {L}}(h(x),y)]=\int _{X\times Y}{\mathcal {L}}(h(x),y)\,d\rho (x,y)} In our setting, we have h = A ( S n ) {\displaystyle h={\mathcal {A}}(S_{n})} , where A {\displaystyle {\mathcal {A}}} is a learning algorithm and S n = ( ( x 1 , y 1 ) , … , ( x n , y n ) ) ∼ ρ n {\displaystyle S_{n}=((x_{1},y_{1}),\ldots ,(x_{n},y_{n}))\sim \rho ^{n}} is a sequence of vectors which are all drawn independently from ρ {\displaystyle \rho } . Define the optimal risk E H ∗ = inf h ∈ H E ( h ) . {\displaystyle {\mathcal {E}}_{\mathcal {H}}^{}={\underset {h\in {\mathcal {H}}}{\inf }}{\mathcal {E}}(h).} Set h n = A ( S n ) {\displaystyle h_{n}={\mathcal {A}}(S_{n})} , for each sample size n {\displaystyle n} . h n {\displaystyle h_{n}} is a random variable and depends on the random variable S n {\displaystyle S_{n}} , which is drawn from the distribution ρ n {\displaystyle \rho ^{n}} . The algorithm A {\displaystyle {\mathcal {A}}} is called consistent if E ( h n ) {\displaystyle {\mathcal {E}}(h_{n})} probabilistically converges to E H ∗ {\displaystyle {\mathcal {E}}_{\mathcal {H}}^{}} . In other words, for all ϵ , δ > 0 {\displaystyle \epsilon ,\delta >0} , there exists a positive integer N {\displaystyle N} , such that, for all sample sizes n ≥ N {\displaystyle n\geq N} , we have Pr ρ n [ E ( h n ) − E H ∗ ≥ ε ] < δ . {\displaystyle \Pr _{\rho ^{n}}[{\mathcal {E}}(h_{n})-{\mathcal {E}}_{\mathcal {H}}^{}\geq \varepsilon ]<\delta .} The sample complexity of A {\displaystyle {\mathcal {A}}} is then the minimum N {\displaystyle N} for which this holds, as a function of ρ , ϵ {\displaystyle \rho ,\epsilon } , and δ {\displaystyle \delta } . We write the sample complexity as N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} to emphasize that this value of N {\displaystyle N} depends on ρ , ϵ {\displaystyle \rho ,\epsilon } , and δ {\displaystyle \delta } . If A {\displaystyle {\mathcal {A}}} is not consistent, then we set N ( ρ , ϵ , δ ) = ∞ {\displaystyle N(\rho ,\epsilon ,\delta )=\infty } . If there exists an algorithm for which N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} is finite, then we say that the hypothesis space H {\displaystyle {\mathcal {H}}} is learnable. In others words, the sample complexity N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} defines the rate of consistency of the algorithm: given a desired accuracy ϵ {\displaystyle \epsilon } and confidence δ {\displaystyle \delta } , one needs to sample N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} data points to guarantee that the risk of the output function is within ϵ {\displaystyle \epsilon } of the best possible, with probability at least 1 − δ {\displaystyle 1-\delta } . In probably approximately correct (PAC) learning, one is concerned with whether the sample complexity is polynomial, that is, whether N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} is bounded by a polynomial in 1 / ϵ {\displaystyle 1/\epsilon } and 1 / δ {\displaystyle 1/\delta } . If N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} is polynomial for some learning algorithm, then one says that the hypothesis space H {\displaystyle {\mathcal {H}}} is PAC-learnable. This is a stronger notion than being learnable. == Unrestricted hypothesis space: infinite sample complexity == One can ask whether there exists a learning algorithm so that the sample complexity is finite in the strong sense, that is, there is a bound on the number of samples needed so that the algorithm can learn any distribution over the input-output space with a specified target error. More formally, one asks whether there exists a learning algorithm A {\displaystyle {\mathcal {A}}} , such that, for all ϵ , δ > 0 {\displaystyle \epsilon ,\delta >0} , there exists a positive integer N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} , we have sup ρ ( Pr ρ n [ E ( h n ) − E H ∗ ≥ ε ] ) < δ , {\displaystyle \sup _{\rho }\left(\Pr _{\rho ^{n}}[{\mathcal {E}}(h_{n})-{\mathcal {E}}_{\mathcal {H}}^{}\geq \varepsilon ]\right)<\delta ,} where h n = A ( S n ) {\displaystyle h_{n}={\mathcal {A}}(S_{n})} , with S n = ( ( x 1 , y 1 ) , … , ( x n , y n ) ) ∼ ρ n {\displaystyle S_{n}=((x_{1},y_{1}),\ldots ,(x_{n},y_{n}))\sim \rho ^{n}} as above. The No Free Lunch Theorem says that without restrictions on the hypothesis space H {\displaystyle {\mathcal {H}}} , this is not the case, i.e., there always exist "bad" distributions for which the sample complexity is arbitrarily large. Thus, in order to make statements about the rate of convergence of the quantity sup ρ ( Pr ρ n [ E ( h n ) − E H ∗ ≥ ε ] ) , {\displaystyle \sup _{\rho }\left(\Pr _{\rho ^{n}}[{\mathcal {E}}(h_{n})-{\mathcal {E}}_{\mathcal {H}}^{}\geq \varepsilon ]\right),} one must either constrain the space of probability distributions ρ {\displaystyle \rho } , e.g. via a parametric approach, or constrain the space of hypotheses H {\displaystyle {\mathcal {H}}} , as in distribution-free approaches. == Restricted hypothesis space: finite sample-complexity == The latter approach leads to concepts such as VC dimension and Rademacher complexity which control the complexity of the space H {\displaystyle {\mathcal {H}}} . A smaller hypothesis space introduces more bias into the inference process, meaning that E H ∗ {\displaystyle {\mathcal {E}}_{\mathcal {H}}^{}} may be greater than the best possible risk in a larger space. However, by restricting the complexity of the hypothesis space it becomes possible for an algorithm to produce more uniformly consistent functions. This trade-off leads to the concept of regularization. It is a theorem from VC theory that the following three statements are equivalent for a hypothesis space H {\displaystyle {\mathcal {H}}} : H {\displaystyle {\mathcal {H}}} is PAC-learnable. The VC dimension of H {\displaystyle {\mathcal {H}}} is finite. H {\displaystyle {\mathcal {H}}} is a uniform Glivenko-Cantelli class. This gives a way to prove that certain hypothesis spaces are PAC learnable, and by extension, learnable. === An example of a PAC-learnable hypothesis space === X = R d , Y = { − 1 , 1 } {\displaystyle X=\mathbb {R} ^{d},Y=\{-1,1\}} , and let H {\displaystyle {\mathcal {H}}} be the space of affine functions on X {\displaystyle X} , that is, functions of the form x ↦ ⟨ w , x ⟩ + b {\displaystyle x\mapsto \langl
EfficientNet
EfficientNet is a family of convolutional neural networks (CNNs) for computer vision published by researchers at Google AI in 2019. Its key innovation is compound scaling, which uniformly scales all dimensions of depth, width, and resolution using a single parameter. EfficientNet models have been adopted in various computer vision tasks, including image classification, object detection, and segmentation. == Compound scaling == EfficientNet introduces compound scaling, which, instead of scaling one dimension of the network at a time, such as depth (number of layers), width (number of channels), or resolution (input image size), uses a compound coefficient ϕ {\displaystyle \phi } to scale all three dimensions simultaneously. Specifically, given a baseline network, the depth, width, and resolution are scaled according to the following equations: depth multiplier: d = α ϕ width multiplier: w = β ϕ resolution multiplier: r = γ ϕ {\displaystyle {\begin{aligned}{\text{depth multiplier: }}d&=\alpha ^{\phi }\\{\text{width multiplier: }}w&=\beta ^{\phi }\\{\text{resolution multiplier: }}r&=\gamma ^{\phi }\end{aligned}}} subject to α ⋅ β 2 ⋅ γ 2 ≈ 2 {\displaystyle \alpha \cdot \beta ^{2}\cdot \gamma ^{2}\approx 2} and α ≥ 1 , β ≥ 1 , γ ≥ 1 {\displaystyle \alpha \geq 1,\beta \geq 1,\gamma \geq 1} . The α ⋅ β 2 ⋅ γ 2 ≈ 2 {\displaystyle \alpha \cdot \beta ^{2}\cdot \gamma ^{2}\approx 2} condition is such that increasing ϕ {\displaystyle \phi } by a factor of ϕ 0 {\displaystyle \phi _{0}} would increase the total FLOPs of running the network on an image approximately 2 ϕ 0 {\displaystyle 2^{\phi _{0}}} times. The hyperparameters α {\displaystyle \alpha } , β {\displaystyle \beta } , and γ {\displaystyle \gamma } are determined by a small grid search. The original paper suggested 1.2, 1.1, and 1.15, respectively. Architecturally, they optimized the choice of modules by neural architecture search (NAS), and found that the inverted bottleneck convolution (which they called MBConv) used in MobileNet worked well. The EfficientNet family is a stack of MBConv layers, with shapes determined by the compound scaling. The original publication consisted of 8 models, from EfficientNet-B0 to EfficientNet-B7, with increasing model size and accuracy. EfficientNet-B0 is the baseline network, and subsequent models are obtained by scaling the baseline network by increasing ϕ {\displaystyle \phi } . == Variants == EfficientNet has been adapted for fast inference on edge TPUs and centralized TPU or GPU clusters by NAS. EfficientNet V2 was published in June 2021. The architecture was improved by further NAS search with more types of convolutional layers. It also introduced a training method, which progressively increases image size during training, and uses regularization techniques like dropout, RandAugment, and Mixup. The authors claim this approach mitigates accuracy drops often associated with progressive resizing.
POP-11
POP-11 is a reflective, incrementally compiled programming language with many of the features of an interpreted language. It is the core language of the Poplog programming environment developed originally by the University of Sussex, and recently in the School of Computer Science at the University of Birmingham, which hosts the main Poplog website. POP-11 is an evolution of the language POP-2, developed in Edinburgh University, and features an open stack model (like Forth, among others). It is mainly procedural, but supports declarative language constructs, including a pattern matcher, and is mostly used for research and teaching in artificial intelligence, although it has features sufficient for many other classes of problems. It is often used to introduce symbolic programming techniques to programmers of more conventional languages like Pascal, who find POP syntax more familiar than that of Lisp. One of POP-11's features is that it supports first-class functions. POP-11 is the core language of the Poplog system. The availability of the compiler and compiler subroutines at run-time (a requirement for incremental compiling) gives it the ability to support a far wider range of extensions (including run-time extensions, such as adding new data-types) than would be possible using only a macro facility. This made it possible for (optional) incremental compilers to be added for Prolog, Common Lisp and Standard ML, which could be added as required to support either mixed language development or development in the second language without using any POP-11 constructs. This made it possible for Poplog to be used by teachers, researchers, and developers who were interested in only one of the languages. The most successful product developed in POP-11 was the Clementine data mining system, developed by ISL. After SPSS bought ISL, they renamed Clementine to SPSS Modeler and decided to port it to C++ and Java, and eventually succeeded with great effort, and perhaps some loss of the flexibility provided by the use of an AI language. POP-11 was for a time available only as part of an expensive commercial package (Poplog), but since about 1999 it has been freely available as part of the open-source software version of Poplog, including various added packages and teaching libraries. An online version of ELIZA using POP-11 is available at Birmingham. At the University of Sussex, David Young used POP-11 in combination with C and Fortran to develop a suite of teaching and interactive development tools for image processing and vision, and has made them available in the Popvision extension to Poplog. == Simple code examples == Here is an example of a simple POP-11 program: define Double(Source) -> Result; Source2 -> Result; enddefine; Double(123) => That prints out: 246 This one includes some list processing: define RemoveElementsMatching(Element, Source) -> Result; lvars Index; [[% for Index in Source do unless Index = Element or Index matches Element then Index; endunless; endfor; %]] -> Result; enddefine; RemoveElementsMatching("the", [[the cat sat on the mat]]) => ;;; outputs [[cat sat on mat]] RemoveElementsMatching("the", [[the cat] [sat on] the mat]) => ;;; outputs [[the cat] [sat on] mat] RemoveElementsMatching([[= cat]], [[the cat]] is a [[big cat]]) => ;;; outputs [[is a]] Examples using the POP-11 pattern matcher, which makes it relatively easy for students to learn to develop sophisticated list-processing programs without having to treat patterns as tree structures accessed by 'head' and 'tail' functions (CAR and CDR in Lisp), can be found in the online introductory tutorial. The matcher is at the heart of the SimAgent (sim_agent) toolkit. Some of the powerful features of the toolkit, such as linking pattern variables to inline code variables, would have been very difficult to implement without the incremental compiler facilities.
Visual descriptor
In computer vision, visual descriptors or image descriptors are descriptions of the visual features of the contents in images, videos, or algorithms or applications that produce such descriptions. They describe elementary characteristics such as the shape, the color, the texture or the motion, among others. == Introduction == As a result of the new communication technologies and the massive use of Internet in our society, the amount of audio-visual information available in digital format is increasing considerably. Therefore, it has been necessary to design some systems that allow us to describe the content of several types of multimedia information in order to search and classify them. The audio-visual descriptors are in charge of the contents description. These descriptors have a good knowledge of the objects and events found in a video, image or audio and they allow the quick and efficient searches of the audio-visual content. This system can be compared to the search engines for textual contents. Although it is relatively easy to find text with a computer, it is much more difficult to find concrete audio and video parts. For instance, imagine somebody searching a scene of a happy person. The happiness is a feeling and it is not evident its shape, color and texture description in images. The description of the audio-visual content is not a superficial task and it is essential for the effective use of this type of archives. The standardization system that deals with audio-visual descriptors is the MPEG-7 (Motion Picture Expert Group - 7). == Types == Descriptors are the first step to find out the connection between pixels contained in a digital image and what humans recall after having observed an image or a group of images after some minutes. Visual descriptors are divided in two main groups: General information descriptors: contain low level descriptors which give a description about color, shape, regions, textures and motion. Specific domain information descriptors: give information about objects and events in the scene. A concrete example would be face recognition. === General information descriptors === General information descriptors consist of a set of descriptors that covers different basic and elementary features like: color, texture, shape, motion, location and others. This description is automatically generated by means of signal processing. ==== Color ==== It's the most basic quality of visual content. Five tools are defined to describe color. The three first tools represent the color distribution and the last ones describe the color relation between sequences or group of images: Dominant color descriptor (DCD) Scalable color descriptor (SCD) Color structure descriptor (CSD) Color layout descriptor (CLD) Group of frame (GoF) or group-of-pictures (GoP) ==== Texture ==== It's an important quality in order to describe an image. The texture descriptors characterize image textures or regions. They observe the region homogeneity and the histograms of these region borders. The set of descriptors is formed by: Homogeneous texture descriptor (HTD) Texture browsing descriptor (TBD) Edge histogram descriptor (EHD) ==== Shape ==== It contains important semantic information due to human's ability to recognize objects through their shape. However, this information can only be extracted by means of a segmentation similar to the one that the human visual system implements. Nowadays, such a segmentation system is not available yet, however there exists a serial of algorithms which are considered to be a good approximation. These descriptors describe regions, contours and shapes for 2D images and for 3D volumes. The shape descriptors are the following ones: Region-based shape descriptor (RSD) Contour-based shape descriptor (CSD) 3-D shape descriptor (3-D SD) ==== Motion ==== It's defined by four different descriptors which describe motion in video sequence. Motion is related to the objects motion in the sequence and to the camera motion. This last information is provided by the capture device, whereas the rest is implemented by means of image processing. The descriptor set is the following one: Motion activity descriptor (MAD) Camera motion descriptor (CMD) Motion trajectory descriptor (MTD) Warping and parametric motion descriptor (WMD and PMD) ==== Location ==== Elements location in the image is used to describe elements in the spatial domain. In addition, elements can also be located in the temporal domain: Region locator descriptor (RLD) Spatio temporal locator descriptor (STLD) === Specific domain information descriptors === These descriptors, which give information about objects and events in the scene, are not easily extractable, even more when the extraction is to be automatically done. Nevertheless, they can be manually processed. As mentioned before, face recognition is a concrete example of an application that tries to automatically obtain this information. == Descriptors applications == Among all applications, the most important ones are: Multimedia documents search engines and classifiers. Digital library: visual descriptors allow a very detailed and concrete search of any video or image by means of different search parameters. For instance, the search of films where a known actor appears, the search of videos containing the Everest mountain, etc. Personalized electronic news service. Possibility of an automatic connection to a TV channel broadcasting a soccer match, for example, whenever a player approaches the goal area. Control and filtering of concrete audiovisual content, like violent or pornographic material. Also, authorization for some multimedia content.
Multimodal representation learning
Multimodal representation learning is a subfield of representation learning focused on integrating and interpreting information from different modalities, such as text, images, audio, or video, by projecting them into a shared latent space. This allows for semantically similar content across modalities to be mapped to nearby points within that space, facilitating a unified understanding of diverse data types. By automatically learning meaningful features from each modality and capturing their inter-modal relationships, multimodal representation learning enables a unified representation that enhances performance in cross-media analysis tasks such as video classification, event detection, and sentiment analysis. It also supports cross-modal retrieval and translation, including image captioning, video description, and text-to-image synthesis. == Motivation == The primary motivations for multimodal representation learning arise from the inherent nature of real-world data and the limitations of unimodal approaches. Since multimodal data offers complementary and supplementary information about an object or event from different perspectives, it is more informative than relying on a single modality. A key motivation is to narrow the heterogeneity gap that exists between different modalities by projecting their features into a shared semantic subspace. This allows semantically similar content across modalities to be represented by similar vectors, facilitating the understanding of relationships and correlations between them. Multimodal representation learning aims to leverage the unique information provided by each modality to achieve a more comprehensive and accurate understanding of concepts. These unified representations are crucial for improving performance in various cross-media analysis tasks such as video classification, event detection, and sentiment analysis. They also enable cross-modal retrieval, allowing users to search and retrieve content across different modalities. Additionally, it facilitates cross-modal translation, where information can be converted from one modality to another, as seen in applications like image captioning and text-to-image synthesis. The abundance of ubiquitous multimodal data in real-world applications, including understudied areas like healthcare, finance, and human-computer interaction (HCI), further motivates the development of effective multimodal representation learning techniques. == Approaches and methods == === Canonical-correlation analysis based methods === Canonical-correlation analysis (CCA) was first introduced in 1936 by Harold Hotelling and is a fundamental approach for multimodal learning. CCA aims to find linear relationships between two sets of variables. Given two data matrices X ∈ R n × p {\displaystyle X\in \mathbb {R} ^{n\times p}} and Y ∈ R n × q {\displaystyle Y\in \mathbb {R} ^{n\times q}} representing different modalities, CCA finds projection vectors w x ∈ R p {\displaystyle w_{x}\in \mathbb {R} ^{p}} and w y ∈ R q {\displaystyle w_{y}\in \mathbb {R} ^{q}} that maximizes the correlation between the projected variables: ρ = max w x , w y w x ⊤ Σ x y w y w x ⊤ Σ x x w x w y ⊤ Σ y y w y {\displaystyle \rho =\max _{w_{x},w_{y}}{\frac {w_{x}^{\top }\Sigma _{xy}w_{y}}{{\sqrt {w_{x}^{\top }\Sigma _{xx}w_{x}}}{\sqrt {w_{y}^{\top }\Sigma _{yy}w_{y}}}}}} such that Σ x x {\displaystyle \Sigma _{xx}} and Σ y y {\displaystyle \Sigma _{yy}} are the within-modality covariance matrices, and Σ x y {\displaystyle \Sigma _{xy}} is the between-modality covariance matrix. However, standard CCA is limited by its linearity, which led to the development of nonlinear extensions, such as kernel CCA and deep CCA. ==== Kernel CCA ==== Kernel canonical correlation analysis (KCCA) extends traditional CCA to capture nonlinear relationships between modalities by implicitly mapping the data into high dimensional feature spaces using kernel functions. Given kernel functions K x {\displaystyle K_{x}} and K y {\displaystyle K_{y}} with corresponding Gram matrices K x ∈ R n × n {\displaystyle K_{x}\in \mathbb {R} ^{n\times n}} and K y ∈ R n × n {\displaystyle K_{y}\in \mathbb {R} ^{n\times n}} , KCCA seeks coefficients α {\displaystyle \alpha } and β {\displaystyle \beta } that maximize: ρ = max α , β α ⊤ K x K y β α ⊤ K x 2 α β ⊤ K y 2 β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{\top }K_{x}Ky\beta }{{\sqrt {\alpha ^{\top }K_{x}^{2}\alpha }}{\sqrt {\beta ^{\top }K_{y}^{2}\beta }}}}} To prevent overfitting, regularization terms are typically added, resulting in: ρ = max α , β α T K x K y β α T ( K x 2 + λ x K x ) α β T ( K y 2 + λ y K y ) β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{T}K_{x}K_{y}\beta }{{\sqrt {\alpha ^{T}\left(K_{x}^{2}+\lambda _{x}K_{x}\right)\alpha }}{\sqrt {\;\beta ^{T}\left(K_{y}^{2}+\lambda _{y}K_{y}\right)\beta }}}}} where λ x {\displaystyle \lambda _{x}} and λ y {\displaystyle \lambda _{y}} are regularization parameters. KCCA has proven effective for tasks such as cross-modal retrieval and semantic analysis, though it faces computational challenges with large datasets due to its O ( n 2 ) {\displaystyle O(n^{2})} memory requirement for sorting kernel matrices. KCCA was proposed independently by several researchers. ==== Deep CCA ==== Deep canonical correlation analysis (DCCA), introduced in 2013, employs neural networks to learn nonlinear transformations for maximizing the correlation between modalities. DCCA uses separate neural networks f x {\displaystyle f_{x}} and f y {\displaystyle f_{y}} for each modality to transform the original data before applying CCA: max W x , W y , θ x , θ y corr ( f x ( X ; θ x ) , f y ( Y ; θ y ) ) {\displaystyle \max _{W_{x},W_{y},\theta _{x},\theta _{y}}\operatorname {corr} \left(f_{x}(X;\theta _{x}),f_{y}(Y;\theta _{y})\right)} where θ x {\displaystyle \theta _{x}} and θ y {\displaystyle \theta _{y}} represent the parameters of the neural networks, and W x {\displaystyle W_{x}} and W y {\displaystyle W_{y}} are the CCA projection matrices. The correlation objective is computed as: corr ( H x , H y ) = tr ( T − 1 / 2 H x T H y S − 1 / 2 ) {\displaystyle \operatorname {corr} (H_{x},H_{y})=\operatorname {tr} \left(T^{-1/2}H_{x}^{T}H_{y}S^{-1/2}\right)} where H x = f x ( X ) {\displaystyle H_{x}=f_{x}(X)} and H y = f y ( Y ) {\displaystyle H_{y}=f_{y}(Y)} are the network outputs, T = H x T H x + r x I {\displaystyle T=H_{x}^{T}H_{x}+r_{x}I} , S = H y T H y + r y I {\displaystyle S=H_{y}^{T}H_{y}+r_{y}I} and r x , r y {\displaystyle r_{x},r_{y}} are the regularization parameters. DCCA overcomes the limitations of linear CCA and kernel CCA by learning complex nonlinear relationships while maintaining computational efficiency for large datasets through mini-batch optimization. === Graph-based methods === Graph-based approaches for multimodal representation learning leverage graph structure to model relationships between entities across different modalities. These methods typically represent each modality as a graph and then learn embedding that preserve cross-modal similarities, enabling more effective joint representation of heterogeneous data. One such method is cross-modal graph neural networks (CMGNNs) that extend traditional graph neural networks (GNNs) to handle data from multiple modalities by constructing graphs that capture both intra-modal and inter-modal relationships. These networks model interactions across modalities by representing them as nodes and their relationships as edges. Other graph-based methods include Probabilistic Graphical Models (PGMs) such as deep belief networks (DBN) and deep Boltzmann machines (DBM). These models can learn a joint representation across modalities, for instance, a multimodal DBN achieves this by adding a shared restricted Boltzmann Machine (RBM) hidden layer on top of modality-specific DBNs. Additionally, the structure of data in some domains like Human-Computer Interaction (HCI), such as the view hierarchy of app screens, can potentially be modeled using graph-like structures. The field of graph representation learning is also relevant, with ongoing progress in developing evaluation benchmarks. === Diffusion maps === Another set of methods relevant to multimodal representation learning are based on diffusion maps and their extensions to handle multiple modalities. ==== Multi-view diffusion maps ==== Multi-view diffusion maps address the challenge of achieving multi-view dimensionality reduction by effectively utilizing the availability of multiple views to extract a coherent low-dimensional representation of the data. The core idea is to exploit both the intrinsic relations within each view and the mutual relations between the different views, defining a cross-view model where a random walk process implicitly hops between objects in different views. A multi-view kernel matrix is constructed by combining these relations, defining a cross-view diffusion process and associ