Manifold regularization

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 ${\displaystyle f}$ from among a hypothesis space of functions ${\displaystyle {\mathcal {H}}}$. The hypothesis space is an RKHS, meaning that it is associated with a kernel ${\displaystyle K}$, and so every candidate function ${\displaystyle f}$ has a norm ${\displaystyle \left\|f\right\|_{K}}$, 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 ${\displaystyle (x_{1},y_{1}),\ldots ,(x_{\ell },y_{\ell })}$ with ${\displaystyle x_{i}\in X,y_{i}\in Y}$ and a loss function ${\displaystyle V}$, a learning algorithm using Tikhonov regularization will attempt to solve the expression

${\displaystyle {\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }V(f(x_{i}),y_{i})+\gamma \left\|f\right\|_{K}^{2}}$

where ${\displaystyle \gamma }$ is a hyperparameter that controls how much the algorithm will prefer simpler functions over functions that fit the data better.

A two-dimensional manifold embedded in three-dimensional space (left). Manifold regularization attempts to learn a function that is smooth on the unrolled manifold (right).

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 ${\displaystyle X}$, but instead from a nonlinear manifold ${\displaystyle M\subset X}$. The geometry of this manifold, the intrinsic space, is used to determine the regularization norm.[1]

There are many possible choices for the intrinsic regularizer ${\displaystyle \left\|f\right\|_{I}}$. Many natural choices involve the gradient on the manifold ${\displaystyle \nabla _{M}}$, 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 ${\displaystyle \nabla _{M}f(x)}$ should be small where the marginal probability density ${\displaystyle {\mathcal {P}}_{X}(x)}$, the probability density of a randomly drawn data point appearing at ${\displaystyle x}$, is large. This gives one appropriate choice for the intrinsic regularizer:

${\displaystyle \left\|f\right\|_{I}^{2}=\int _{x\in M}\left\|\nabla _{M}f(x)\right\|^{2}\,d{\mathcal {P}}_{X}(x)}$

In practice, this norm cannot be computed directly because the marginal distribution ${\displaystyle {\mathcal {P}}_{X}}$ is unknown, but it can be estimated from the provided data.

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 ${\displaystyle \ell }$ labeled examples (pairs of an input ${\displaystyle x}$ and a label ${\displaystyle y}$) and ${\displaystyle u}$ unlabeled examples (inputs without associated labels). Define ${\displaystyle W}$ to be a matrix of edge weights for a graph, where ${\displaystyle W_{ij}}$ is a distance measure between the data points ${\displaystyle x_{i}}$ and ${\displaystyle x_{j}}$. Define ${\displaystyle D}$ to be a diagonal matrix with ${\displaystyle D_{ii}=\sum _{j=1}^{\ell +u}W_{ij}}$ and ${\displaystyle L}$ to be the Laplacian matrix ${\displaystyle D-W}$. Then, as the number of data points ${\displaystyle \ell +u}$ increases, ${\displaystyle L}$ converges to the Laplace–Beltrami operator ${\displaystyle \Delta _{M}}$, which is the divergence of the gradient ${\displaystyle \nabla _{M}}$.[2][3] Then, if ${\displaystyle \mathbf {f} }$ is a vector of the values of ${\displaystyle f}$ at the data, ${\displaystyle \mathbf {f} =[f(x_{1}),\ldots ,f(x_{l+u})]^{\mathrm {T} }}$, the intrinsic norm can be estimated:

${\displaystyle \left\|f\right\|_{I}^{2}={\frac {1}{(\ell +u)^{2}}}\mathbf {f} ^{\mathrm {T} }L\mathbf {f} }$

As the number of data points ${\displaystyle \ell +u}$ increases, this empirical definition of ${\displaystyle \left\|f\right\|_{I}^{2}}$ converges to the definition when ${\displaystyle {\mathcal {P}}_{X}}$ is known.[1]

Using the weights ${\displaystyle \gamma _{A}}$ and ${\displaystyle \gamma _{I}}$ for the ambient and intrinsic regularizers, the final expression to be solved becomes:

${\displaystyle {\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }V(f(x_{i}),y_{i})+\gamma _{A}\left\|f\right\|_{K}^{2}+{\frac {\gamma _{I}}{(\ell +u)^{2}}}\mathbf {f} ^{\mathrm {T} }L\mathbf {f} }$

As with other kernel methods, ${\displaystyle {\mathcal {H}}}$ 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 ${\displaystyle \left\|f\right\|_{I}}$, the optimal solution ${\displaystyle f^{*}}$ must be a linear combination of the kernel centered at each of the input points: for some weights ${\displaystyle \alpha _{i}}$,

${\displaystyle f^{*}(x)=\sum _{i=1}^{\ell +u}\alpha _{i}K(x_{i},x)}$

Using this result, it is possible to search for the optimal solution ${\displaystyle f^{*}}$ by searching the finite-dimensional space defined by the possible choices of ${\displaystyle \alpha _{i}}$.[1]

The idea beyond graph-Laplacian is to use neighbors to estimate Laplacian. This method is akin local averaging methods, that are known to scale poorly in high-dimensional problem. Indeed, graph Laplacian is known to suffer from the curse of dimensionality.[2] Luckily, it is possible to leverage expected smoothness of the function to estimate thanks to more advanced functional analysis. This method consists in estimating the Laplacian operator thanks to derivatives of the kernel reading ${\displaystyle \partial _{1,j}K(x_{i},x)}$ where ${\displaystyle \partial _{1,j}}$ denotes the partial derivatives according to the j-th coordinate of the first variable.[4] This second approach of the Laplacian norm is to put in relation with meshfree methods, that contrast with the finite difference method in PDE.

Regularized least squares (RLS) is a family of regression algorithms: algorithms that predict a value ${\displaystyle y=f(x)}$ for its inputs ${\displaystyle x}$, with the goal that the predicted values should be close to the true labels for the data. In particular, RLS is designed to minimize the mean squared error between the predicted values and the true labels, subject to regularization. Ridge regression is one form of RLS; in general, RLS is the same as ridge regression combined with the kernel method. The problem statement for RLS results from choosing the loss function ${\displaystyle V}$ in Tikhonov regularization to be the mean squared error:

${\displaystyle f^{*}={\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }(f(x_{i})-y_{i})^{2}+\gamma \left\|f\right\|_{K}^{2}}$

Thanks to the representer theorem, the solution can be written as a weighted sum of the kernel evaluated at the data points:

${\displaystyle f^{*}(x)=\sum _{i=1}^{\ell }\alpha _{i}^{*}K(x_{i},x)}$

and solving for ${\displaystyle \alpha ^{*}}$ gives:

${\displaystyle \alpha ^{*}=(K+\gamma \ell I)^{-1}Y}$

where ${\displaystyle K}$ is defined to be the kernel matrix, with ${\displaystyle K_{ij}=K(x_{i},x_{j})}$, and ${\displaystyle Y}$ is the vector of data labels.

Adding a Laplacian term for manifold regularization gives the Laplacian RLS statement:

${\displaystyle f^{*}={\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }(f(x_{i})-y_{i})^{2}+\gamma _{A}\left\|f\right\|_{K}^{2}+{\frac {\gamma _{I}}{(\ell +u)^{2}}}\mathbf {f} ^{\mathrm {T} }L\mathbf {f} }$

The representer theorem for manifold regularization again gives

${\displaystyle f^{*}(x)=\sum _{i=1}^{\ell +u}\alpha _{i}^{*}K(x_{i},x)}$

and this yields an expression for the vector ${\displaystyle \alpha ^{*}}$. Letting ${\displaystyle K}$ be the kernel matrix as above, ${\displaystyle Y}$ be the vector of data labels, and ${\displaystyle J}$ be the ${\displaystyle (\ell +u)\times (\ell +u)}$ block matrix ${\displaystyle {\begin{bmatrix}I_{\ell }&0\\0&0_{u}\end{bmatrix}}}$:

${\displaystyle \alpha ^{*}={\underset {\alpha \in \mathbf {R} ^{\ell +u}}{\arg \!\min }}{\frac {1}{\ell }}(Y-JK\alpha )^{\mathrm {T} }(Y-JK\alpha )+\gamma _{A}\alpha ^{\mathrm {T} }K\alpha +{\frac {\gamma _{I}}{(\ell +u)^{2}}}\alpha ^{\mathrm {T} }KLK\alpha }$

with a solution of

${\displaystyle \alpha ^{*}=\left(JK+\gamma _{A}\ell I+{\frac {\gamma _{I}\ell }{(\ell +u)^{2}}}LK\right)^{-1}Y}$[1]

LapRLS has been applied to problems including sensor networks,[7] medical imaging,[8][9] object detection,[10] spectroscopy,[11] document classification,[12] drug-protein interactions,[13] and compressing images and videos.[14]

Support vector machines (SVMs) are a family of algorithms often used for classifying data into two or more groups, or classes. Intuitively, an SVM draws a boundary between classes so that the closest labeled examples to the boundary are as far away as possible. This can be directly expressed as a linear program, but it is also equivalent to Tikhonov regularization with the hinge loss function, ${\displaystyle V(f(x),y)=\max(0,1-yf(x))}$:

${\displaystyle f^{*}={\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }\max(0,1-y_{i}f(x_{i}))+\gamma \left\|f\right\|_{K}^{2}}$[15][16]

Adding the intrinsic regularization term to this expression gives the LapSVM problem statement:

${\displaystyle f^{*}={\underset {f\in {\mathcal {H}}}{\arg \!\min }}{\frac {1}{\ell }}\sum _{i=1}^{\ell }\max(0,1-y_{i}f(x_{i}))+\gamma _{A}\left\|f\right\|_{K}^{2}+{\frac {\gamma _{I}}{(\ell +u)^{2}}}\mathbf {f} ^{\mathrm {T} }L\mathbf {f} }$

Again, the representer theorem allows the solution to be expressed in terms of the kernel evaluated at the data points:

${\displaystyle f^{*}(x)=\sum _{i=1}^{\ell +u}\alpha _{i}^{*}K(x_{i},x)}$

${\displaystyle \alpha }$ can be found by writing the problem as a linear program and solving the dual problem. Again letting ${\displaystyle K}$ be the kernel matrix and ${\displaystyle J}$ be the block matrix ${\displaystyle {\begin{bmatrix}I_{\ell }&0\\0&0_{u}\end{bmatrix}}}$, the solution can be shown to be

${\displaystyle \alpha =\left(2\gamma _{A}I+2{\frac {\gamma _{I}}{(\ell +u)^{2}}}LK\right)^{-1}J^{\mathrm {T} }Y\beta ^{*}}$

where ${\displaystyle \beta ^{*}}$ is the solution to the dual problem

${\displaystyle {\begin{aligned}&&\beta ^{*}=\max _{\beta \in \mathbf {R} ^{\ell }}&\sum _{i=1}^{\ell }\beta _{i}-{\frac {1}{2}}\beta ^{\mathrm {T} }Q\beta \\&{\text{subject to}}&&\sum _{i=1}^{\ell }\beta _{i}y_{i}=0\\&&&0\leq \beta _{i}\leq {\frac {1}{\ell }}\;i=1,\ldots ,\ell \end{aligned}}}$

and ${\displaystyle Q}$ is defined by

${\displaystyle Q=YJK\left(2\gamma _{A}I+2{\frac {\gamma _{I}}{(\ell +u)^{2}}}LK\right)^{-1}J^{\mathrm {T} }Y}$[1]

LapSVM has been applied to problems including geographical imaging,[17][18][19] medical imaging,[20][21][22] face recognition,[23] machine maintenance,[24] and brain–computer interfaces.[25]

• The ManifoldLearn library and the Primal LapSVM library implement LapRLS and LapSVM in MATLAB.
• The Dlib library for C++ includes a linear manifold regularization function.