Perform DDRTree construction

This is an R and C code implementation of the DDRTree algorithm from Qi Mao, Li Wang et al.

Qi Mao, Li Wang, Steve Goodison, and Yijun Sun. Dimensionality Reduction via Graph Structure Learning.
The 21st ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD'15), 2015

http://dl.acm.org/citation.cfm?id=2783309

to perform dimension reduction and principal graph
learning simultaneously. Please cite this package and KDD'15 paper if you found DDRTree is useful for your research.

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`X` |
a matrix with |

`dimensions` |
reduced dimension |

`maxIter` |
maximum iterations |

`sigma` |
bandwidth parameter |

`lambda` |
regularization parameter for inverse graph embedding |

`ncenter` |
number of nodes allowed in the regularization graph |

`param.gamma` |
regularization parameter for k-means (the prefix of 'param' is used to avoid name collision with gamma) |

`tol` |
relative objective difference |

`verbose` |
emit extensive debug output |

a list with W, Z, stree, Y, history W is the orthogonal set of d (dimensions) linear basis vector Z is the reduced dimension space stree is the smooth tree graph embedded in the low dimension space Y represents latent points as the center of Z

The unprecedented increase in big-data causes a huge difficulty in data visualization and downstream analysis.
Conventional dimension reduction approaches (for example, PCA, ICA, Isomap, LLE, etc.) are limited in their ability
to explictly recover the intrinisic structure from the data as well as the discriminative feature representation,
both are important for scientific discovery. The DDRTree algorithm is a new algorithm to perform the following
three tasks in one setting:

1. Reduce high dimension data into a low dimension space

2. Recover an explicit smooth graph structure with local geometry only captured by distances of data
points in the low dimension space.

3. Obtain clustering structures of data points in reduced dimension

Reverse graph embedding is previously applied to learn the intrinisic graph structure in the original dimension. The optimization of graph inference can be represented as:

*\mathop{min}_{f_g \in \mathcal{F}} \mathop{min}_{\{\mathbf{z}_1, ..., \mathbf{z}_M\}} ∑_{(V_i, V_j) \in
\mathcal{E}} b_{i,j}||f_g(\mathbf{z}_i) - f_g(\mathbf{z}_j)||^2*

where
*f_g* is a function to map the instrinsic data space *\mathcal{Z} = \{\mathbf{z}_1, ..., \mathbf{z}_M\}* back to the input data space (reverse embedding)
*\mathcal{X} = \{ \mathbf{x}_1, ..., \mathbf{x}_N\}*. *V_i* is the the vertex of the instrinsic undirected graph
*\mathcal{G} = (\mathcal{V}, \mathcal{E})*. *b_{ij}* is
the edge weight associates with the edge set *\mathcal{E}*.
In order to learn the intrinsic structure from a reduced dimension, we need also to consider a term which includes
the error during the learning of the instrinsic structure. This strategy is incorporated as the following:

*\mathop{min}_{\mathcal{G} \in \hat{\mathcal{G}}_b}\mathop{min}_{f_g \in \mathcal{F}} \mathop{min}_{\{\mathbf{z}_1, ...,
\mathbf{z}_M\}} ∑_{i = 1}^N ||\mathbf{x}_i - f_g (\mathbf{z}_i)||^2 + \frac{λ}{2} ∑_{(V_i, V_j) \in \mathcal{E}}
b_{i,j}||f_g(\mathbf{z}_i) - f_g(\mathbf{z}_j)||^2*

where *λ* is a non-negative parameter which controls the tradeoff between the data
reconstruction error and the reverse graph embedding.

The general framework for reducing dimension by learning an intrinsic structure in a low dimension requires a feasible set
*\hat{\mathcal{G}}_b* of graph and a mapping function *f_\mathcal{G}*. The algorithm uses minimum spanning tree as the feasible tree
graph structure, which can be solved by Kruskal' algoritm. A linear projection model *f_g (\mathbf{z}) = \mathbf{Wz}* is used as the mapping function.
Those setting results in the following specific form for the previous framework:

*\mathop{min}_{\mathbf{W}, \mathbf{Z}, \mathbf{B}} ∑_{i = 1}^N ||\mathbf{x}_i - \mathbf{W}\mathbf{z}_i||^2 + \frac{λ}{2} ∑_{i,j}b_{i,j}||\mathbf{W} \mathbf{z}_i - \mathbf{W} \mathbf{z}_j||^2*

where *\mathbf{W} = [\mathbf{w}_1, ..., \mathbf{w}_d] \in
\mathcal{R}^{D \times d}* is an orthogonal set of *d* linear basis vectors. We can group tree graph *\mathbf{B}*, the orthogonal set of linear basis vectors
and projected points in reduced dimension *\mathbf{W}, \mathbf{Z}* as two groups and apply alternative structure optimization to optimize the tree graph.
This method is defined as DRtree (Dimension Reduction tree) as discussed by the authors.

In order to avoid the issues where data points scattered into different branches (which leads to lose of cluster information) and to
incorporate the discriminative information,another set of points *\{\mathbf{y}_k\}_{k = 1}^K* as the centers of *\{\mathbf{z}_i\}^N_{i = 1}* can be also introduced.
By so doing, the objective functions of K-means and the DRtree can be simulatenously minimized. The author further proposed a soft partition method
to account for the limits from K-means and proposed the following objective function:

*\mathop{min}_{\mathbf{W}, \mathbf{Z}, \mathbf{B}, \mathbf{Y}, \mathbf{R}} ∑_{i = 1}^N ||\mathbf{x}_i - \mathbf{W} \mathbf{z}_i||^2 +
\frac{λ}{2} ∑_{k, k'}b_{k, k'}||\mathbf{W} \mathbf{y}_k - \mathbf{W} \mathbf{y}_k'||^2 +
γ\Big[∑_{k = 1}^K ∑_{i = 1}^N r_{i, k} ||\mathbf{z}_i - \mathbf{y}_k||^2 + σ Ω (\mathbf{R})\Big]*

*s.t.\ \mathbf{W}^T \mathbf{W} = \mathbf{I}, \mathbf{B} \in \mathcal{B}, ∑_{k = 1}^K r_{i, k} = 1,
r_{i, k} ≤q 0, \forall i, \forall k*

where *\mathbf{R} \in \mathcal{R}^{N \times N}, Ω(\mathbf{R}) = ∑_{i = 1}^N ∑_{k = 1}^k r_{i, k} log\ r_{i, k}* is the negative
entropy regularization which transforms the hard assignments used in K-means into soft assignments and *σ > 0* is the regulization parameter.
Alternative structure optimization is again used to solve the above problem by separately optimize each group *{\mathbf{W}, \mathbf{Z}, \mathbf{Y}}, {\mathbf{B}, \mathbf{R}}* until convergence.

*1.* *\mathbf{Input}*: Data matrix *\mathbf{X}*, parameters *λ, σ, γ*

*2.* Initialize *\mathbf{Z}* by PCA

*3.* *K = N, \mathbf{Y} = \mathbf{Z}*

*4.* *\mathbf{repeat}*:

*\ 5.* *d_{k,k'} = ||\mathbf{y}_k - \mathbf{y}_{k'}||^2, \forall k, \forall k'*

*\ 6.* Obtain *\mathbf{B}* via Kruskal's algorithm

*\ 7.* *\mathbf{L} = diag(\mathbf{B1}) - \mathbf{B}*

*\ 8.* Compute *\mathbf{R}* with each element

*\ 9.* *τ = diag(\mathbf{1}^T \mathbf{R})*

*\ 10.* *\mathbf{Q} = \frac{1}{\mathbf{1} + γ} \Big[\mathbf{I} + \mathbf{R} (\frac{1 + γ}{γ}(\frac{λ}{γ} \mathbf{L} +
τ) - \mathbf{R}^T \mathbf{R})^{-1} \mathbf{R}^T\Big]*

*\ 11.* *\mathbf{C} = \mathbf{X Q X}^T*

*\ 12.* Perform eigen-decomposition on *\mathbf{C}* such that *\mathbf{C} = \mathbf{U} \wedge \mathbf{U}^T* and *diag(\wedge)* is sorted in a descending order

*\ 13.* *\mathbf{W} = \mathbf{U}(:, 1:d)*

*\ 14.* *\mathbf{Z} = \mathbf{W}^T \mathbf{X Q}*

*\ 15.* *\mathbf{Y} = \mathbf{Z R} (\frac{λ}{γ} \mathbf{L} + τ)^{-1}*

*16.* *\mathbf{Until}* Convergence

We implemented the algorithm mostly in Rcpp for the purpose of efficiency. It also has extensive optimization for sparse input data. This implementation is originally based on the matlab code provided from the author of DDRTree paper.

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data('iris')
subset_iris_mat <- as.matrix(t(iris[c(1, 2, 52, 103), 1:4])) #subset the data
#run DDRTree with ncenters equal to species number
DDRTree_res <- DDRTree(subset_iris_mat, dimensions = 2, maxIter = 5, sigma = 1e-2,
lambda = 1, ncenter = 3, param.gamma = 10, tol = 1e-2, verbose = FALSE)
Z <- DDRTree_res$Z #obatain matrix
Y <- DDRTree_res$Y
stree <- DDRTree_res$stree
plot(Z[1, ], Z[2, ], col = iris[c(1, 2, 52, 103), 'Species']) #reduced dimension
legend("center", legend = unique(iris[c(1, 2, 52, 103), 'Species']), cex=0.8,
col=unique(iris[c(1, 2, 52, 103), 'Species']), pch = 1) #legend
title(main="DDRTree reduced dimension", col.main="red", font.main=4)
dev.off()
plot(Y[1, ], Y[2, ], col = 'blue', pch = 17) #center of the Z
title(main="DDRTree smooth principal curves", col.main="red", font.main=4)
#run DDRTree with ncenters equal to species number
DDRTree_res <- DDRTree(subset_iris_mat, dimensions = 2, maxIter = 5, sigma = 1e-3,
lambda = 1, ncenter = NULL,param.gamma = 10, tol = 1e-2, verbose = FALSE)
Z <- DDRTree_res$Z #obatain matrix
Y <- DDRTree_res$Y
stree <- DDRTree_res$stree
plot(Z[1, ], Z[2, ], col = iris[c(1, 2, 52, 103), 'Species']) #reduced dimension
legend("center", legend = unique(iris[c(1, 2, 52, 103), 'Species']), cex=0.8,
col=unique(iris[c(1, 2, 52, 103), 'Species']), pch = 1) #legend
title(main="DDRTree reduced dimension", col.main="red", font.main=4)
dev.off()
plot(Y[1, ], Y[2, ], col = 'blue', pch = 2) #center of the Z
title(main="DDRTree smooth principal graphs", col.main="red", font.main=4)
``` |

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