Vegas Pro (formerly known as Sony Vegas) is a professional video editing software package for non-linear editing (NLE), designed to run on the Microsoft Windows operating system. The first release of Vegas Beta was on June 11, 1999. Vegas was originally developed as a non-linear audio editing application. Version 2.0 would split the program into audio and video editing variants, with the former being dropped by version 4.0, making the video offering the only variant available to consumers. Vegas Pro features real-time multi-track video and audio editing on unlimited tracks, resolution-independent video sequencing, complex effects, compositing tools, 24-bit/192 kHz audio support, VST and DirectX plug-in effect support, and Dolby Digital surround sound mixing. The software was originally published by Sonic Foundry until May 2003, when Sony purchased Sonic Foundry and formed Sony Creative Software. On May 24, 2016, Sony announced that Vegas was sold to MAGIX, which formed VEGAS Creative Software, to continue support and development of the software. As of the end of March 2026, it was publicly announced that Boris FX had taken ownership of Vegas Pro. Each release of Vegas is sold standalone; however, upgrade discounts are sometimes provided. == Features == Vegas does not require any specialized hardware to run properly, allowing it to operate on any Windows computer that meets the system requirements. == History == Vegas 1.0 was released after a brief public beta by Sonic Foundry on July 23, 1999 at the NAMM Show in Nashville, Tennessee as an audio-only tool with a particular focus on re-scaling and resampling audio. It supported formats like DivX and Real Networks RealSystem G2 file formats. Martin Walker from Sound on Sound described working in Vegas 1.0 as a "very pleasurable experience, especially since so many functions are highly intuitive" though also criticizing some features as hard to figure out due to the lack of a central help file. Later, on June 12, 2000, Vegas Video and Audio 2.0 (also referred to as just Vegas 2.0) was released, with its beta releasing earlier that year on April 10. This was the first version of Vegas to include video-editing tools and was also the first to have a low-cost "LE" version alongside the regular release. The LE releases would continue through version 3.0 of Vegas but would be discontinued by the release of Vegas 4.0. Vegas 3.0 was released the next year on December 3, and added new video effects, features for ease-of-use with DV, and support for editing Windows Media files. Vegas 4.0 was released on 6 February 2003 and added application scripting, advanced color correction, 5.1 surround sound mixing, and Steinberg ASIO support. This was the last release under the Sonic Foundry name after it sold much of its software suite, including Sound Forge and Acid Pro, to Sony Pictures Digital for $18 million later in 2003. Under Sony's ownership, Vegas 5.0 was released on April 19, 2004, bringing 3D track motion, compositing, reversing, envelope automation, etc. 7.0 also added an improved video preview, enhanced layout management, improved snapping, and more customization. With the release of 8.0, Sony opted to go back to the original "Vegas Pro" branding that the first version released with. It added the ability to burn Blu-ray and DVD optical media, support for 32-bit floating point audio, support for tempo-based audio effects, and more. It also moved the timeline to the bottom of the window by default with the option of moving it back to the top if the user wished to. Sony was also experimenting with 64-bit at this time and ported Vegas Pro 8.0 to 64-bit systems under the name "Vegas Pro 8.1". Vegas Pro 9.0 added support for 4K resolution and pro camcorder formats like Red and XDCAM EX. In 2009, Sony Creative Software purchased the Velvetmatter Radiance suite of video FX plug-ins which were included in Sony Vegas Pro 9.0. As a result, they were no longer available as a separate product from Velvetmatter. Vegas Pro 10 was released in 2010 with stereoscopic 3D editing, image stabilization, OpenFX plugin support, real-time audio event effects, and a few UI changes. This was the last release to include support for Windows XP. Vegas Pro 11 was released the next year on 17 October, with GPGPU video acceleration, enhanced text tools, enhanced stereoscopic/3D features, RAW photo support, and new event synchronization mechanisms. In addition, Vegas Pro 11 comes pre-loaded with "NewBlue" Titler Pro, a 2D and 3D titling plug-in. Vegas Pro 12 would add two new configurations: Vegas Pro 12 Edit, for "Professional Video and Audio Production"; and Vegas Pro 12 Suite, for "Professional Editing, Disc Authoring, and Visual Effects Design". Vegas Pro 13 would be the last version released with Sony branding after the acquisition of much of Sony Creative Software's library by Magix. After they acquired Vegas, Magix released version 14 on September 20, 2016. It featured advanced 4K upscaling as well as many bug fixes, a higher video velocity limit, RED camera support, and a variety of other features. This was also the last version to have the light theme enabled by default. Released on August 28, 2017, Vegas Pro 15 features major UI changes that claim to bring usability improvements and customization. It was the first version of VEGAS Pro to have a dark theme; it also allows more efficient editing speeds, including adding new shortcuts to speed the video editing process. Vegas Pro 15 includes support for Intel Quick Sync Video (QSV) and other technologies, as well as various other features. It introduced a new VEGAS Pro icon as a V. Vegas Pro 16 has some new features including file backup, motion tracking, improved video stabilization, 360° editing and HDR support. Magix has continued to improve Vegas through version 21 with support for reading Matroska files, a more detailed render dialogue, live streaming, VST3 support, a VST 32-bit bridge, and a selective Paste Event Attributes menu. Magix would later release a subscription model for using Vegas named "Vegas Pro 365" on January 17, 2018, although the perpetual licence is still an option for customers. This version includes cloud-based speech synthesis among other features not included in the mainline Vegas release. == Version history == Each release of Vegas is sold standalone, however upgrade discounts are sometimes provided. === Vegas Beta === Sonic Foundry introduced a sneak preview version of Vegas Pro on June 11, 1999. It is called a "Multitrack Media Editing System". === Vegas 1.0 === Released on July 23, 1999 at the NAMM Show in Nashville, Tennessee, Vegas was an audio-only tool with a particular focus on rescaling and resampling audio. It supported formats like DivX and Real Networks RealSystem G2 file formats. Version 1.0 is the final Vegas release to include Windows 95 support. === Vegas Video beta (Vegas 2.0 beta) === Released on April 10, 2000, this was the first version of Vegas to include video-editing tools. === Vegas Video (Vegas 2.0) === Released on June 12, 2000. Version 2.0 is the final Vegas Video release to include Windows NT 4.0 support. === Vegas Video 3.0 === Released on December 3, 2001. This release added: New Video Effects – Lens Flare, Light Rays, Film FX, Color Curves, Mirror, Remap, Deform, Convolution, Linear Blur, Black Restore, Levels, Unsharp Mask, Color Grading, and Timecode Burn filter. Batch Capture with Automatic Scene Detection – Captures DV with automatic scene detection, batch capture, tape logging, still image capture and thumbnail previews. Red Book Audio CD Mastering with CD Architect (TM) Technology – Used for burning Red Book audio CD masters directly from the Vegas timeline with ISRC, UPC, and PQ list support. New Sonic Foundry DV Codec – Introduces a DV codec developed by Sonic Foundry that offers artifact-free compositing and DV chromakeying. DV Print-to-Tape from the Timeline – Prints projects to DV cameras and decks from the Vegas timeline. Windows Media (TM) File Editing – Creates and edits Windows Media (TM) files. New MPEG Encoding Tools – Used for producing MPEG-2 files for DVD productions. Dynamic RAM Previewing – Temporary RAM/render-free previews for analysis and tweaking of complex video FX without rendering. VideoCD and Data CD Burning – Burning projects directly to VideoCD for playback on most DVD players or data CDs for playback computers' CD-ROMs. === Vegas 4.0 === Released on February 6, 2003. This release added: Advanced Color Correction Tools Searchable Media Pool Bins Vectorscope, Histogram, Parade and Waveform Monitoring Application Scripting Improved Ripple Editing Motion Blur and Super-Sampling Envelopes 5.1 Surround Mixing Dolby® Digital AC-3 Encoding certified and tested by Dolby Laboratories DirectX® Audio Plug-In Effects Automation ASIO Driver Support Windows Media™ 9 Support, including Surround Encoding DVD Authoring with AC-3 File Import Capabilities Integration with DVD Architect via Chap
Adversarial stylometry
Adversarial stylometry is the practice of altering writing style to reduce the potential for stylometry to discover the author's identity or their characteristics. This task is also known as authorship obfuscation or authorship anonymisation. Stylometry poses a significant privacy challenge in its ability to unmask anonymous authors or to link pseudonyms to an author's other identities, which, for example, creates difficulties for whistleblowers, activists, and hoaxers and fraudsters. The privacy risk is expected to grow as machine learning techniques and text corpora develop. All adversarial stylometry shares the core idea of faithfully paraphrasing the source text so that the meaning is unchanged but the stylistic signals are obscured. Such a faithful paraphrase is an adversarial example for a stylometric classifier. Several broad approaches to this exist, with some overlap: imitation, substituting the author's own style for another's; translation, applying machine translation with the hope that this eliminates characteristic style in the source text; and obfuscation, deliberately modifying a text's style to make it not resemble the author's own. Manually obscuring style is possible, but laborious; in some circumstances, it is preferable or necessary. Automated tooling, either semi- or fully-automatic, could assist an author. How best to perform the task and the design of such tools is an open research question. While some approaches have been shown to be able to defeat particular stylometric analyses, particularly those that do not account for the potential of adversariality, establishing safety in the face of unknown analyses is an issue. Ensuring the faithfulness of the paraphrase is a critical challenge for automated tools. It is uncertain if the practice of adversarial stylometry is detectable in itself. Some studies have found that particular methods produced signals in the output text, but a stylometrist who is uncertain of what methods may have been used may not be able to reliably detect them. == History == Rao & Rohatgi (2000), an early work in adversarial stylometry, identified machine translation as a possibility, but noted that the quality of translators available at the time presented severe challenges. Kacmarcik & Gamon (2006) is another early work. Brennan, Afroz & Greenstadt (2012) performed the first evaluation of adversarial stylometric methods on actual texts. Brennan & Greenstadt (2009) introduced the first corpus of adversarially authored texts specifically for evaluating stylometric methods; other corpora include the International Imitation Hemingway Competition, the Faux Faulkner contest, and the hoax blog A Gay Girl in Damascus. == Motivations == Rao & Rohatgi (2000) suggest that short, unattributed documents (i.e., anonymous posts) are not at risk of stylometric identification, but pseudonymous authors who have not practiced adversarial stylometry in producing corpuses of thousands of words may be vulnerable. Narayanan et al. (2012) attempted large-scale deanonymisation of 100,000 blog authors with mixed results: the identifications were significantly better than chance, but only accurately matched the blog and author a fifth of the time; identification improved with the number of posts written by the author in the corpus. Even if an author is not identified, some of their characteristics may still be deduced stylometrically, or stylometry may narrow the anonymity set of potential authors sufficiently for other information to complete the identification. Detecting author characteristics (e.g., gender or age) is often simpler than identifying an author from a large, possibly open, set of candidates. Modern machine learning techniques offer powerful tools for identification; further development of corpora and computational stylometric techniques are likely to raise further privacy issues. Gröndahl & Asokan (2020a) say that the general validity of the hypothesis underlying stylometry—that authors have invariant, content-independent 'style fingerprints'—is uncertain, but "the deanonymisation attack is a real privacy concern". Those interested in practicing adversarial stylometry and stylistic deception include whistleblowers avoiding retribution; journalists and activists; perpetrators of frauds and hoaxes; authors of fake reviews; literary forgers; criminals disguising their identity from investigators; and, generally, anyone with a desire for anonymity or pseudonymity. Authors, or agents acting on behalf of authors, may also attempt to remove stylistic clues to author characteristics (e.g., race or gender) so that knowledge of those characteristics cannot be used for discrimination (e.g., through algorithmic bias). Another possible use for adversarial stylometry is in disguising automatically generated text as human-authored. == Methods == With imitation, the author attempts to mislead stylometry by matching their style to another author's. An incomplete imitation, where some of the true author's unique characteristics appear alongside the imitated author's, can be a detectable signal for the use of adversarial stylometry. Imitation can be performed automatically with style transfer systems, though this typically requires a large corpus in the target style for the system to learn from. Another approach is translation, which employs machine translation of a source text to eliminate characteristic style, often through multiple translators in sequence to produce a round-trip translation. Such chained translation can lead to texts being significantly altered, even to the point of incomprehensibility; improved translation tools reduce this risk. More simply-structured texts can be easier to machine translate without losing the original meaning. Machine translation blurs into direct stylistic imitation or obfuscation achieved through automated style transfer, which can be viewed as a "translation" with the same language as input and output. With low-quality translation tools, an author can be required to manually correct major translation errors while avoiding the hazard of re-introducing stylistic characteristics. Wang, Juola & Riddell (2022) found that gross errors introduced by Google Translate were rare, but more common with several intermediate translations—however, occasional simple or short sentences and misspellings in the source text appeared verbatim in the output, potentially providing an identifying signal. Chain translation can leave characteristic traces of its application in a document, which may allow reconstruction of the intermediate languages used and the number of translation steps performed. Obfuscation involves deliberately changing the style of a text to reduce its similarity to other texts by some metric; this may be performed at the time of writing by conscious modification, or as part of a revision process with feedback from the metric being targeted as an input to decide when the text has been sufficiently obfuscated. In contrast to translation, complex texts can offer more opportunities for effective obfuscation without altering meaning, and likewise genres with more permissible variation allow more obfuscation. However, longer texts are harder to thoroughly obfuscate. Obfuscation can blend into imitation if the author develops a novel target style, distinct from their original style. With respect to masking author characteristics, obfuscation may aim to achieve a union (adding signals for imitated characteristics) or an intersection (removing signals and normalising) of other authors' styles. Avoiding the author's own idiosyncrasies and producing a "normalised" text is a critical obfuscatory step: an author may have a unique tendency to misspell certain words, use particular variants, or to format a document in a characteristic way. Stylometric signals vary in how simply they can be adversarially masked; an author may easily change their vocabulary by conscious choice, but altering the pattern of grammar or the letter frequency in their text may be harder to achieve, though Juola & Vescovi (2011) report that imitation typically succeeds at masking more characteristics than obfuscation. Automated obfuscation may require large amounts of training data written by the author. Concerning automated implementations of adversarial stylometry, two possible implementations are rule-based systems for paraphrasing; and encoder–decoder architectures, where the text passes through an intermediate format that is (intended to be) style-neutral. Another division in automated methods is whether there is feedback from an identification system or not. With such feedback, finding paraphrases for author masking has been characterised as a heuristic search problem, exploring textual variants until the result is stylistically sufficiently far (in the case of obfuscation) or near (in the case of imitation), which then constitutes an adversarial example for that identification system. == Evaluation == How
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Steve Omohundro
Stephen Malvern Omohundro (born 1959) is an American computer scientist whose areas of research include Hamiltonian physics, dynamical systems, programming languages, machine learning, machine vision, and the social implications of artificial intelligence. His current work uses rational economics to develop safe and beneficial intelligent technologies for better collaborative modeling, understanding, innovation, and decision making. == Education == Omohundro has degrees in physics and mathematics from Stanford University (Phi Beta Kappa) and a Ph.D. in physics from the University of California, Berkeley. == Learning algorithms == Omohundro started the "Vision and Learning Group" at the University of Illinois, which produced 4 Masters and 2 Ph.D. theses. His work in learning algorithms included a number of efficient geometric algorithms, the manifold learning task and various algorithms for accomplishing this task, other related visual learning and modelling tasks, the best-first model merging approach to machine learning (including the learning of Hidden Markov Models and Stochastic Context-free Grammars), and the Family Discovery Learning Algorithm, which discovers the dimension and structure of a parameterized family of stochastic models. == Self-improving artificial intelligence and AI safety == Omohundro started Self-Aware Systems in Palo Alto, California to research the technology and social implications of self-improving artificial intelligence. He is an advisor to the Machine Intelligence Research Institute on artificial intelligence. He argues that rational systems exhibit problematic natural "drives" that will need to be countered in order to build intelligent systems safely. His papers, talks, and videos on AI safety have generated extensive interest. He has given many talks on self-improving artificial intelligence, cooperative technology, AI safety, and connections with biological intelligence. == Programming languages == At Thinking Machines Corporation, Cliff Lasser and Steve Omohundro developed Star Lisp, the first programming language for the Connection Machine. Omohundro joined the International Computer Science Institute (ICSI) in Berkeley, California, where he led the development of the open source programming language Sather. Sather is featured in O'Reilly's History of Programming Languages poster. == Physics and dynamical systems theory == Omohundro's book Geometric Perturbation Theory in Physics describes natural Hamiltonian symplectic structures for a wide range of physical models that arise from perturbation theory analyses. He showed that there exist smooth partial differential equations which stably perform universal computation by simulating arbitrary cellular automata. The asymptotic behavior of these PDEs is therefore logically undecidable. With John David Crawford he showed that the orbits of three-dimensional period doubling systems can form an infinite number of topologically distinct torus knots and described the structure of their stable and unstable manifolds. == Mathematica and Apple tablet contest == From 1986 to 1988, he was an Assistant Professor of Computer science at the University of Illinois at Urbana-Champaign and cofounded the Center for Complex Systems Research with Stephen Wolfram and Norman Packard. While at the University of Illinois, he worked with Stephen Wolfram and five others to create the symbolic mathematics program Mathematica. He and Wolfram led a team of students that won an Apple Computer contest to design "The Computer of the Year 2000." Their design entry "Tablet" was a touchscreen tablet with GPS and other features that finally appeared when the Apple iPad was introduced 22 years later. == Other contributions == Subutai Ahmad and Steve Omohundro developed biologically realistic neural models of selective attention. As a research scientist at the NEC Research Institute, Omohundro worked on machine learning and computer vision, and was a co-inventor of U.S. Patent 5,696,964, "Multimedia Database Retrieval System Which Maintains a Posterior Probability Distribution that Each Item in the Database is a Target of a Search." === Pirate puzzle === Omohundro developed an extension to the game theoretic pirate puzzle featured in Scientific American. == Outreach == Omohundro has sat on the Machine Intelligence Research Institute board of advisors. He has written extensively on artificial intelligence, and has warned that "an autonomous weapons arms race is already taking place" because "military and economic pressures are driving the rapid development of autonomous systems".
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Manifold regularization
In machine learning, manifold regularization is a technique for using the shape of a dataset to constrain the functions that should be learned on that dataset. In many machine learning problems, the data to be learned do not cover the entire input space. For example, a facial recognition system may not need to classify any possible image, but only the subset of images that contain faces. The technique of manifold learning assumes that the relevant subset of data comes from a manifold, a mathematical structure with useful properties. The technique also assumes that the function to be learned is smooth: data with different labels are not likely to be close together, and so the labeling function should not change quickly in areas where there are likely to be many data points. Because of this assumption, a manifold regularization algorithm can use unlabeled data to inform where the learned function is allowed to change quickly and where it is not, using an extension of the technique of Tikhonov regularization. Manifold regularization algorithms can extend supervised learning algorithms in semi-supervised learning and transductive learning settings, where unlabeled data are available. The technique has been used for applications including medical imaging, geographical imaging, and object recognition. == Manifold regularizer == === Motivation === Manifold regularization is a type of regularization, a family of techniques that reduces overfitting and ensures that a problem is well-posed by penalizing complex solutions. In particular, manifold regularization extends the technique of Tikhonov regularization as applied to Reproducing kernel Hilbert spaces (RKHSs). Under standard Tikhonov regularization on RKHSs, a learning algorithm attempts to learn a function f {\displaystyle f} from among a hypothesis space of functions H {\displaystyle {\mathcal {H}}} . The hypothesis space is an RKHS, meaning that it is associated with a kernel K {\displaystyle K} , and so every candidate function f {\displaystyle f} has a norm ‖ f ‖ K {\displaystyle \left\|f\right\|_{K}} , which represents the complexity of the candidate function in the hypothesis space. When the algorithm considers a candidate function, it takes its norm into account in order to penalize complex functions. Formally, given a set of labeled training data ( x 1 , y 1 ) , … , ( x ℓ , y ℓ ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{\ell },y_{\ell })} with x i ∈ X , y i ∈ Y {\displaystyle x_{i}\in X,y_{i}\in Y} and a loss function V {\displaystyle V} , a learning algorithm using Tikhonov regularization will attempt to solve the expression arg min f ∈ H 1 ℓ ∑ i = 1 ℓ V ( f ( x i ) , y i ) + γ ‖ f ‖ K 2 {\displaystyle {\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }V(f(x_{i}),y_{i})+\gamma \left\|f\right\|_{K}^{2}} where γ {\displaystyle \gamma } is a hyperparameter that controls how much the algorithm will prefer simpler functions over functions that fit the data better. Manifold regularization adds a second regularization term, the intrinsic regularizer, to the ambient regularizer used in standard Tikhonov regularization. Under the manifold assumption in machine learning, the data in question do not come from the entire input space X {\displaystyle X} , but instead from a nonlinear manifold M ⊂ X {\displaystyle M\subset X} . The geometry of this manifold, the intrinsic space, is used to determine the regularization norm. === Laplacian norm === There are many possible choices for the intrinsic regularizer ‖ f ‖ I {\displaystyle \left\|f\right\|_{I}} . Many natural choices involve the gradient on the manifold ∇ M {\displaystyle \nabla _{M}} , which can provide a measure of how smooth a target function is. A smooth function should change slowly where the input data are dense; that is, the gradient ∇ M f ( x ) {\displaystyle \nabla _{M}f(x)} should be small where the marginal probability density P X ( x ) {\displaystyle {\mathcal {P}}_{X}(x)} , the probability density of a randomly drawn data point appearing at x {\displaystyle x} , is large. This gives one appropriate choice for the intrinsic regularizer: ‖ f ‖ I 2 = ∫ x ∈ M ‖ ∇ M f ( x ) ‖ 2 d P X ( x ) {\displaystyle \left\|f\right\|_{I}^{2}=\int _{x\in M}\left\|\nabla _{M}f(x)\right\|^{2}\,d{\mathcal {P}}_{X}(x)} In practice, this norm cannot be computed directly because the marginal distribution P X {\displaystyle {\mathcal {P}}_{X}} is unknown, but it can be estimated from the provided data. === Graph-based approach of the Laplacian norm === When the distances between input points are interpreted as a graph, then the Laplacian matrix of the graph can help to estimate the marginal distribution. Suppose that the input data include ℓ {\displaystyle \ell } labeled examples (pairs of an input x {\displaystyle x} and a label y {\displaystyle y} ) and u {\displaystyle u} unlabeled examples (inputs without associated labels). Define W {\displaystyle W} to be a matrix of edge weights for a graph, where W i j {\displaystyle W_{ij}} is a similarity built from distance measure between the data points x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} (so that more close implies higher W i j {\displaystyle W_{ij}} ). Define D {\displaystyle D} to be a diagonal matrix with D i i = ∑ j = 1 ℓ + u W i j {\displaystyle D_{ii}=\sum _{j=1}^{\ell +u}W_{ij}} and L {\displaystyle L} to be the Laplacian matrix D − W {\displaystyle D-W} . Then, as the number of data points ℓ + u {\displaystyle \ell +u} increases, L {\displaystyle L} converges to the Laplace–Beltrami operator Δ M {\displaystyle \Delta _{M}} , which is the divergence of the gradient ∇ M {\displaystyle \nabla _{M}} . Then, if f {\displaystyle \mathbf {f} } is a vector of the values of f {\displaystyle f} at the data, f = [ f ( x 1 ) , … , f ( x l + u ) ] T {\displaystyle \mathbf {f} =[f(x_{1}),\ldots ,f(x_{l+u})]^{\mathrm {T} }} , the intrinsic norm can be estimated: ‖ f ‖ I 2 = 1 ( ℓ + u ) 2 f T L f {\displaystyle \left\|f\right\|_{I}^{2}={\frac {1}{(\ell +u)^{2}}}\mathbf {f} ^{\mathrm {T} }L\mathbf {f} } As the number of data points ℓ + u {\displaystyle \ell +u} increases, this empirical definition of ‖ f ‖ I 2 {\displaystyle \left\|f\right\|_{I}^{2}} converges to the definition when P X {\displaystyle {\mathcal {P}}_{X}} is known. === Solving the regularization problem with graph-based approach === Using the weights γ A {\displaystyle \gamma _{A}} and γ I {\displaystyle \gamma _{I}} for the ambient and intrinsic regularizers, the final expression to be solved becomes: arg min f ∈ H 1 ℓ ∑ i = 1 ℓ V ( f ( x i ) , y i ) + γ A ‖ f ‖ K 2 + γ I ( ℓ + u ) 2 f T L f {\displaystyle {\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }V(f(x_{i}),y_{i})+\gamma _{A}\left\|f\right\|_{K}^{2}+{\frac {\gamma _{I}}{(\ell +u)^{2}}}\mathbf {f} ^{\mathrm {T} }L\mathbf {f} } As with other kernel methods, H {\displaystyle {\mathcal {H}}} may be an infinite-dimensional space, so if the regularization expression cannot be solved explicitly, it is impossible to search the entire space for a solution. Instead, a representer theorem shows that under certain conditions on the choice of the norm ‖ f ‖ I {\displaystyle \left\|f\right\|_{I}} , the optimal solution f ∗ {\displaystyle f^{}} must be a linear combination of the kernel centered at each of the input points: for some weights α i {\displaystyle \alpha _{i}} , f ∗ ( x ) = ∑ i = 1 ℓ + u α i K ( x i , x ) {\displaystyle f^{}(x)=\sum _{i=1}^{\ell +u}\alpha _{i}K(x_{i},x)} Using this result, it is possible to search for the optimal solution f ∗ {\displaystyle f^{}} by searching the finite-dimensional space defined by the possible choices of α i {\displaystyle \alpha _{i}} . === Functional approach of the Laplacian norm === The idea beyond the graph-Laplacian is to use neighbors to estimate the Laplacian. This method is akin to local averaging methods, that are known to scale poorly in high-dimensional problems. Indeed, the graph Laplacian is known to suffer from the curse of dimensionality. Luckily, it is possible to leverage expected smoothness of the function to estimate thanks to more advanced functional analysis. This method consists of estimating the Laplacian operator using derivatives of the kernel reading ∂ 1 , j K ( x i , x ) {\displaystyle \partial _{1,j}K(x_{i},x)} where ∂ 1 , j {\displaystyle \partial _{1,j}} denotes the partial derivatives according to the j-th coordinate of the first variable. This second approach to the Laplacian norm is to put in relation with meshfree methods, that contrast with the finite difference method in PDE. == Applications == Manifold regularization can extend a variety of algorithms that can be expressed using Tikhonov regularization, by choosing an appropriate loss function V {\displaystyle V} and hypothesis space H {\displaystyle {\mathcal {H}}} . Two commonly used examples are the families of support vector machines and regularized least squares algorithm
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