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Manifold alignment

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Manifold alignment is a class of <a href="Machine_learning" title="Machine learning">machine learning</a> algorithms that produce projections between sets of data, given that the original data sets lie on a common <a href="Manifold_learning" class="mw-redirect" title="Manifold learning">manifold</a>. The concept was first introduced as such by Ham, Lee, and Saul in 2003,<a href="#cite_note-1">[1]</a> adding a manifold constraint to the general problem of correlating sets of high-dimensional vectors.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

Manifold alignment assumes that disparate data sets produced by similar generating processes will share a similar underlying <a href="Manifold" title="Manifold">manifold</a> representation. By learning projections from each original space to the shared manifold, correspondences are recovered and knowledge from one domain can be transferred to another. Most manifold alignment techniques consider only two data sets, but the concept extends to arbitrarily many initial data sets.

Consider the case of aligning two data sets, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X_{i}\in \mathbb {R} ^{m}}</annotation>

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</math><img src="./2eee279ac6484a34df2a48bd2ac1cdd25ccf23a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.918ex; height:2.676ex;" alt="{\displaystyle X_{i}\in \mathbb {R} ^{m}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{i}\in \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle Y_{i}\in \mathbb {R} ^{n}}</annotation>

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Manifold alignment algorithms attempt to project both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> into a new d-dimensional space such that the projections both minimize distance between corresponding points and preserve the local manifold structure of the original data. The projection functions are denoted:

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<annotation encoding="application/x-tex">{\displaystyle \phi _{X}:\,\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{d}}</annotation>

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</math><img src="./930356be647ebf7f72ae8a4e234992beaf113d6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.079ex; height:3.009ex;" alt="{\displaystyle \phi _{X}:\,\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{d}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{Y}:\,\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{d}}</annotation>

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</math><img src="./c0a6e0ae51cb77ce353639a3f0907c1f572c3140.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.477ex; height:3.009ex;" alt="{\displaystyle \phi _{Y}:\,\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{d}}" loading="lazy">

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> represent the binary correspondence matrix between points in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle W_{i,j}={\begin{cases}1&if\,X_{i}\leftrightarrow Y_{j}\\0&otherwise\end{cases}}}</annotation>

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</math><img src="./b6de53516e828327617638f6e0c49b98c990c529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.273ex; height:6.176ex;" alt="{\displaystyle W_{i,j}={\begin{cases}1&if\,X_{i}\leftrightarrow Y_{j}\\0&otherwise\end{cases}}}" loading="lazy">

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{X}}</annotation>

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</math><img src="./8f586e8531565307eb464a30ec0917a6b7c91c26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.057ex; height:2.509ex;" alt="{\displaystyle S_{X}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{Y}}">

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<annotation encoding="application/x-tex">{\displaystyle S_{Y}}</annotation>

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</math><img src="./84b711d7cf71796a78fdd0abd4e79079507bf53e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.911ex; height:2.509ex;" alt="{\displaystyle S_{Y}}" loading="lazy"> represent pointwise similarities within data sets. This is usually encoded as the <a href="Heat_kernel" title="Heat kernel">heat kernel</a> of the <a href="Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of a <a href="Nearest_neighbor_graph" title="Nearest neighbor graph">k-nearest neighbor graph</a>.

Finally, introduce a coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \mu \leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq \mu \leq 1}</annotation>

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</math><img src="./6ae63761a22f1cc602266b8ca498251611f3c7c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.924ex; height:2.676ex;" alt="{\displaystyle 0\leq \mu \leq 1}" loading="lazy">, which can be tuned to adjust the weight of the 'preserve manifold structure' goal, versus the 'minimize corresponding point distances' goal.

With these definitions in place, the <a href="Loss_function" title="Loss function">loss function</a> for manifold alignment can be written:

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<annotation encoding="application/x-tex">{\displaystyle \arg \min _{\phi _{X},\phi _{Y}}\mu \sum _{i,j}\left\Vert \phi _{X}\left(X_{i}\right)-\phi _{X}\left(X_{j}\right)\right\Vert ^{2}S_{X,i,j}+\mu \sum _{i,j}\left\Vert \phi _{Y}\left(Y_{i}\right)-\phi _{Y}\left(Y_{j}\right)\right\Vert ^{2}S_{Y,i,j}+\left(1-\mu \right)\sum _{i,j}\Vert \phi _{X}\left(X_{i}\right)-\phi _{Y}\left(Y_{j}\right)\Vert ^{2}W_{i,j}}</annotation>

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</math><img src="./cd6d454e2c1886491b1450f8502c6a8ed7a061c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:116.257ex; height:5.843ex;" alt="{\displaystyle \arg \min _{\phi _{X},\phi _{Y}}\mu \sum _{i,j}\left\Vert \phi _{X}\left(X_{i}\right)-\phi _{X}\left(X_{j}\right)\right\Vert ^{2}S_{X,i,j}+\mu \sum _{i,j}\left\Vert \phi _{Y}\left(Y_{i}\right)-\phi _{Y}\left(Y_{j}\right)\right\Vert ^{2}S_{Y,i,j}+\left(1-\mu \right)\sum _{i,j}\Vert \phi _{X}\left(X_{i}\right)-\phi _{Y}\left(Y_{j}\right)\Vert ^{2}W_{i,j}}" loading="lazy">

Solving this optimization problem is equivalent to solving a <a href="Generalized_eigenvalue_problem" class="mw-redirect" title="Generalized eigenvalue problem">generalized eigenvalue problem</a> using the <a href="Laplacian_matrix" title="Laplacian matrix">graph laplacian</a><a href="#cite_note-3">[3]</a> of the joint matrix, G:

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<annotation encoding="application/x-tex">{\displaystyle G=\left[{\begin{array}{cc}\mu S_{X}&\left(1-\mu \right)W\\\left(1-\mu \right)W^{T}&\mu S_{Y}\end{array}}\right]}</annotation>

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</math><img src="./26ec94061a7496495fb24d40185ab61cadb555b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.374ex; margin-bottom: -0.297ex; width:31.988ex; height:6.509ex;" alt="{\displaystyle G=\left[{\begin{array}{cc}\mu S_{X}&\left(1-\mu \right)W\\\left(1-\mu \right)W^{T}&\mu S_{Y}\end{array}}\right]}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Inter-data_correspondences">Inter-data correspondences</h2></div>

The algorithm described above requires full pairwise correspondence information between input data sets; a <a href="Supervised_learning" title="Supervised learning">supervised learning</a> paradigm. However, this information is usually difficult or impossible to obtain in real world applications. Recent work has extended the core manifold alignment algorithm to <a href="Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">semi-supervised</a>

, <a href="Unsupervised_learning" title="Unsupervised learning">unsupervised</a>

, and <a href="Multiple-instance_learning" class="mw-redirect" title="Multiple-instance learning">multiple-instance</a>

settings.

<div class="mw-heading mw-heading2"><h2 id="One-step_vs._two-step_alignment">One-step vs. two-step alignment</h2></div>

The algorithm described above performs a "one-step" alignment, finding embeddings for both data sets at the same time. A similar effect can also be achieved with "two-step" alignments

, following a slightly modified procedure:

<ol><li>Project each input data set to a lower-dimensional space independently, using any of a variety of <a href="Nonlinear_dimensionality_reduction" title="Nonlinear dimensionality reduction">dimension reduction</a> algorithms.</li>

<li>Perform linear manifold alignment on the embedded data, holding the first data set fixed, mapping each additional data set onto the first's manifold. This approach has the benefit of decomposing the required computation, which lowers memory overhead and allows parallel implementations.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Instance-level_vs._feature-level_projections">Instance-level vs. feature-level projections</h2></div>

Manifold alignment can be used to find linear (feature-level) projections, or nonlinear (instance-level) embeddings. While the instance-level version generally produces more accurate alignments, it sacrifices a great degree of flexibility as the learned embedding is often difficult to parameterize. Feature-level projections allow any new instances to be easily embedded in the manifold space, and projections may be combined to form direct mappings between the original data representations. These properties are especially important for knowledge-transfer applications.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Manifold alignment is suited to problems with several corpora that lie on a shared manifold, even when each corpus is of a different dimensionality. Many real-world problems fit this description, but traditional techniques are not able to take advantage of all corpora at the same time. Manifold alignment also facilitates <a href="Transfer_learning" title="Transfer learning">transfer learning</a>, in which knowledge of one domain is used to jump-start learning in correlated domains.

Applications of manifold alignment include:

<ul><li><a href="Cross-language_information_retrieval" title="Cross-language information retrieval">Cross-language information retrieval</a> / automatic translation<a href="#cite_note-procrustes-8">[8]</a>

<ul><li>By representing documents as vector of word counts, manifold alignment can recover the mapping between documents of different languages.</li>

<li>Cross-language document correspondence is relatively easy to obtain, especially from multi-lingual organizations like the <a href="European_Union" title="European Union">European Union</a>.</li></ul></li>

<li>Transfer learning of policy and state representations for reinforcement learning<a href="#cite_note-procrustes-8">[8]</a></li>

<li>Alignment of <a href="Nuclear_magnetic_resonance_spectroscopy_of_proteins" title="Nuclear magnetic resonance spectroscopy of proteins">protein NMR</a> structures<a href="#cite_note-procrustes-8">[8]</a></li>

<li>Accelerating model learning in robotics by sharing data generated by other robots <a href="#cite_note-9">[9]</a></li></ul>

<ul><li><a href="Manifold_hypothesis" title="Manifold hypothesis">Manifold hypothesis</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFHamDaniel_D._LeeLawrence_K._Saul2003" class="citation journal cs1">Ham, Ji Hun; Daniel D. Lee; Lawrence K. Saul (2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220309062452/ftp://ftp.cis.upenn.edu/pub/datamining/public_html/ReadingGroup/papers/corr_icmlws03.pdf">"Learning high dimensional correspondences from low dimensional manifolds"</a> (PDF). Proceedings of the Twentieth International Conference on Machine Learning (ICML-2003). Archived from <a rel="nofollow" class="external text" href="ftp://ftp.cis.upenn.edu/pub/datamining/public_html/ReadingGroup/papers/corr_icmlws03.pdf">the original</a> (PDF) on 2022-03-09.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFHotelling1936" class="citation journal cs1">Hotelling, H (1936). <a rel="nofollow" class="external text" href="http://cbio.ensmp.fr/~jvert/svn/bibli/local/Hotelling1936Relation.pdf">"Relations between two sets of variates"</a> (PDF). Biometrika. 28 (3–4): 321–377. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2333955">10.2307/2333955</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2333955">2333955</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFXiongF._WangC._Zhang2007" class="citation conference cs1">Xiong, L.; F. Wang; C. Zhang (2007). "Semi-definite manifold alignment". Proceedings of the 18th European Conference on Machine Learning. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.91.7346">10.1.1.91.7346</a>.</cite></li>

<li><cite id="CITEREFWangSridhar_Mahadevan2009" class="citation journal cs1">Wang, Chang; Sridhar Mahadevan (2009). <a rel="nofollow" class="external text" href="https://people.cs.umass.edu/~mahadeva/papers/aaai2009.pdf">"A General Framework for Manifold Alignment"</a> (PDF). AAAI Fall Symposium on Manifold Learning and Its Applications.</cite></li>

<li><cite id="CITEREFWangSridhar_Mahadevan2010" class="citation journal cs1">Wang, Chang; Sridhar Mahadevan (2010). <a rel="nofollow" class="external text" href="http://www.cs.umass.edu/~mahadeva/papers/UM-CS-2010-049.pdf">"Multiscale Manifold Alignment"</a> (PDF). Univ. Of Massachusetts TR UM-CS-2010-049.</cite></li>

<li><cite id="CITEREFMa2012" class="citation book cs1">Ma, Yunqian (Apr 15, 2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LjeGZwEACAAJ&q=Manifold+Learning:+Theory+and+Applications">Manifold Learning Theory and Applications</a>. Taylor & Francis Group. p. 376. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4398-7109-6</bdi>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://www.cs.umass.edu/~chwang/home/ma-html">Chang Wang's Manifold alignment overview</a></li></ul></div><div><div class="zim-footer">

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