@@ -130,8 +130,10 @@ distances between all points. Isomap can be performed with the object
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:align: center
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:scale: 50
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- Complexity
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- ----------
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+ |details-start |
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+ **Complexity **
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+ |details-split |
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+
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The Isomap algorithm comprises three stages:
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1. **Nearest neighbor search. ** Isomap uses
@@ -162,6 +164,8 @@ The overall complexity of Isomap is
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* :math: `k` : number of nearest neighbors
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* :math: `d` : output dimension
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+ |details-end |
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+
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.. topic :: References:
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* `"A global geometric framework for nonlinear dimensionality reduction"
@@ -187,8 +191,9 @@ Locally linear embedding can be performed with function
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:align: center
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:scale: 50
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- Complexity
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- ----------
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+ |details-start |
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+ **Complexity **
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+ |details-split |
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The standard LLE algorithm comprises three stages:
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@@ -209,6 +214,8 @@ The overall complexity of standard LLE is
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* :math: `k` : number of nearest neighbors
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* :math: `d` : output dimension
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+ |details-end |
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+
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.. topic :: References:
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* `"Nonlinear dimensionality reduction by locally linear embedding"
@@ -241,8 +248,9 @@ It requires ``n_neighbors > n_components``.
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:align: center
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:scale: 50
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- Complexity
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- ----------
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+ |details-start |
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+ **Complexity **
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+ |details-split |
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The MLLE algorithm comprises three stages:
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@@ -265,6 +273,8 @@ The overall complexity of MLLE is
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* :math: `k` : number of nearest neighbors
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* :math: `d` : output dimension
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+ |details-end |
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+
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.. topic :: References:
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* `"MLLE: Modified Locally Linear Embedding Using Multiple Weights"
@@ -291,8 +301,9 @@ It requires ``n_neighbors > n_components * (n_components + 3) / 2``.
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:align: center
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:scale: 50
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- Complexity
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- ----------
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+ |details-start |
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+ **Complexity **
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+ |details-split |
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The HLLE algorithm comprises three stages:
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@@ -313,6 +324,8 @@ The overall complexity of standard HLLE is
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* :math: `k` : number of nearest neighbors
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* :math: `d` : output dimension
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+ |details-end |
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+
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.. topic :: References:
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* `"Hessian Eigenmaps: Locally linear embedding techniques for
@@ -335,8 +348,9 @@ preserving local distances. Spectral embedding can be performed with the
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function :func: `spectral_embedding ` or its object-oriented counterpart
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:class: `SpectralEmbedding `.
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- Complexity
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- ----------
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+ |details-start |
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+ **Complexity **
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+ |details-split |
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The Spectral Embedding (Laplacian Eigenmaps) algorithm comprises three stages:
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@@ -358,6 +372,8 @@ The overall complexity of spectral embedding is
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* :math: `k` : number of nearest neighbors
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* :math: `d` : output dimension
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+ |details-end |
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+
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.. topic :: References:
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* `"Laplacian Eigenmaps for Dimensionality Reduction
@@ -383,8 +399,9 @@ tangent spaces to learn the embedding. LTSA can be performed with function
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:align: center
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:scale: 50
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- Complexity
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- ----------
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+ |details-start |
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+ **Complexity **
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+ |details-split |
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The LTSA algorithm comprises three stages:
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@@ -404,6 +421,8 @@ The overall complexity of standard LTSA is
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* :math: `k` : number of nearest neighbors
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* :math: `d` : output dimension
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+ |details-end |
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+
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.. topic :: References:
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* :arxiv: `"Principal manifolds and nonlinear dimensionality reduction via
@@ -448,8 +467,9 @@ the similarities chosen in some optimal ways. The objective, called the
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stress, is then defined by :math: `\sum _{i < j} d_{ij}(X) - \hat {d}_{ij}(X)`
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- Metric MDS
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- ----------
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+ |details-start |
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+ **Metric MDS **
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+ |details-split |
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The simplest metric :class: `MDS ` model, called *absolute MDS *, disparities are defined by
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:math: `\hat {d}_{ij} = S_{ij}`. With absolute MDS, the value :math: `S_{ij}`
@@ -458,8 +478,11 @@ should then correspond exactly to the distance between point :math:`i` and
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Most commonly, disparities are set to :math: `\hat {d}_{ij} = b S_{ij}`.
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- Nonmetric MDS
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- -------------
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+ |details-end |
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+
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+ |details-start |
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+ **Nonmetric MDS **
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+ |details-split |
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Non metric :class: `MDS ` focuses on the ordination of the data. If
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:math: `S_{ij} > S_{jk}`, then the embedding should enforce :math: `d_{ij} <
@@ -490,6 +513,7 @@ in the metric case.
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:align: center
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:scale: 60
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+ |details-end |
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.. topic :: References:
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@@ -551,8 +575,10 @@ The disadvantages to using t-SNE are roughly:
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:align: center
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:scale: 50
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- Optimizing t-SNE
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- ----------------
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+ |details-start |
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+ **Optimizing t-SNE **
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+ |details-split |
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+
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The main purpose of t-SNE is visualization of high-dimensional data. Hence,
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it works best when the data will be embedded on two or three dimensions.
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@@ -601,8 +627,11 @@ but less accurate results.
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provides a good discussion of the effects of the various parameters, as well
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as interactive plots to explore the effects of different parameters.
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- Barnes-Hut t-SNE
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- ----------------
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+ |details-end |
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+
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+ |details-start |
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+ **Barnes-Hut t-SNE **
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+ |details-split |
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The Barnes-Hut t-SNE that has been implemented here is usually much slower than
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other manifold learning algorithms. The optimization is quite difficult
@@ -638,6 +667,7 @@ imply that the data cannot be correctly classified by a supervised model. It
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might be the case that 2 dimensions are not high enough to accurately represent
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the internal structure of the data.
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+ |details-end |
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.. topic :: References:
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