TriMap

TriMap is a dimensionality reduction method that uses triplet constraints to form a low-dimensional embedding of a set of points. The triplet constraints are of the form "point i is closer to point j than point k". The triplets are sampled from the high-dimensional representation of the points and a weighting scheme is used to reflect the importance of each triplet.

TriMap provides a significantly better global view of the data than the other dimensionality reduction methods such t-SNE, LargeVis, and UMAP. The global structure includes relative distances of the clusters, multiple scales in the data, and the existence of possible outliers. We define a global score to quantify the quality of an embedding in reflecting the global structure of the data.

CIFAR-10 dataset (test set) passed through a CNN (n = 10,000, d = 1024): Notice the semantic structure unveiled by TriMap.

The following implementation is in Python. Further details and more experimental results are available in the paper. See the example colab for some analysis.

News!

[Mar 16, 2022] An example colab using TriMap JAX implementation is now available at https://github.com/eamid/examples/blob/master/TriMap.ipynb. We analyze the results on S-curve, MNIST, Fashion MNIST, etc. using t-SNE, UMAP, TriMap, and PCA.

[Feb 17, 2022] A JAX implementation is now available at https://github.com/google-research/google-research/tree/master/trimap. More updates are coming soon!

How to use TriMap

TriMap has a transformer API similar to other sklearn libraries. To use TriMap with the default parameters, simply do:

import trimap
from sklearn.datasets import load_digits

digits = load_digits()

embedding = trimap.TRIMAP().fit_transform(digits.data)

To find the embedding using a precomputed pairwise distance matrix D, pass D as input and set use_dist_matrix to True:

embedding = trimap.TRIMAP(use_dist_matrix=True).fit_transform(D)

You can also pass the precomputed k-nearest neighbors and their corresponding distances as a tuple (knn_nbrs, knn_distances). Note that the rows must be in order, starting from point 0 to n-1. This feature also requires X to compute the embedding

embedding = trimap.TRIMAP(knn_tuple=(knn_nbrs, knn_distances)).fit_transform(X)

To calculate the global score, do:

gs = trimap.TRIMAP(verbose=False).global_score(digits.data, embedding)
print("global score %2.2f" % gs)

Parameters

The list of parameters is given blow:

• n_dims: Number of dimensions of the embedding (default = 2)
• n_inliers: Number of nearest neighbors for forming the nearest neighbor triplets (default = 12).
• n_outliers: Number of outliers for forming the nearest neighbor triplets (default = 4).
• n_random: Number of random triplets per point (default = 3).
• distance: Distance measure ('euclidean' (default), 'manhattan', 'angular' (or 'cosine'), 'hamming')
• weight_temp: Temperature of the logarithm applied to the weights. Larger temperatures generate more compact embeddings. weight_temp=0. corresponds to no transformation (default=0.5).
• weight_adj (deprecated): The value of gamma for the log-transformation (default = 500.0).
• lr: Learning rate (default = 0.1).
• n_iters: Number of iterations (default = 400).

The other parameters include:

• knn_tuple: Use the precomputed nearest-neighbors information in form of a tuple (knn_nbrs, knn_distances) (default = None)
• use_dist_matrix: Use the precomputed pairwise distance matrix (default = False)
• apply_pca: Reduce the number of dimensions of the data to 100 if necessary before applying the nearest-neighbor search (default = True).
• opt_method: Optimization method {'sd' (steepest descent), 'momentum' (GD with momentum), 'dbd' (delta-bar-delta, default)}.
• verbose: Print the progress report (default = False).
• return_seq: Store the intermediate results and return the results in a tensor (default = False).

An example of adjusting these parameters:

import trimap
from sklearn.datasets import load_digits

digits = load_digits()

embedding = trimap.TRIMAP(n_inliers=20,
                          n_outliers=10,
                          n_random=10).fit_transform(digits.data)

The nearest-neighbor calculation is performed using ANNOY.

Examples

The following are some of the results on real-world datasets. The values of nearest-neighbor accuracy and global score are shown as a pair (NN, GS) on top of each figure. For more results, please refer to our paper.

USPS Handwritten Digits (n = 11,000, d = 256)

20 News Groups (n = 18,846, d = 100)

Tabula Muris (n = 53,760, d = 23,433)

MNIST Handwritten Digits (n = 70,000, d = 784)

Fashion MNIST (n = 70,000, d = 784)

TV News (n = 129,685, d = 100)

Runtime of t-SNE, LargeVis, UMAP, and TriMap in the hh:mm:ss format on a single machine with 2.6 GHz Intel Core i5 CPU and 16 GB of memory is given in the following table. We limit the runtime of each method to 12 hours. Also, UMAP runs out of memory on datasets larger than ~4M points.

Installing

Requirements:

• numpy
• scikit-learn
• numba
• annoy

Installing annoy

If you are having trouble with installing annoy on macOS using the command:

pip3 install annoy

you can alternatively try:

pip3 install git+https://github.com/sutao/annoy.git@master

Install Options

If you have all the requirements installed, you can use pip:

sudo pip install trimap

Please regularly check for updates and make sure you are using the most recent version. If you have TriMap installed and would like to upgrade to the newer version, you can use the command:

sudo pip install --upgrade --force-reinstall trimap

An alternative is to install the dependencies manually using anaconda and using pip to install TriMap:

conda install numpy
conda install scikit-learn
conda install numba
conda install annoy
pip install trimap

For a manual install get this package:

wget https://github.com/eamid/trimap/archive/master.zip
unzip master.zip
rm master.zip
cd trimap-master

Install the requirements

sudo pip install -r requirements.txt

or

conda install scikit-learn numba annoy

Install the package

python setup.py install

Support and Contribution

This implementation is still a work in progress. Any comments/suggestions/bug-reports are highly appreciated. Please feel free contact me at: [email protected]. If you would like to contribute to the code, please fork the project and send me a pull request.

Citation

If you use TriMap in your publications, please cite our current reference on arXiv:

@article{2019TRIMAP,
     author = {{Amid}, Ehsan and {Warmuth}, Manfred K.},
     title = "{TriMap: Large-scale Dimensionality Reduction Using Triplets}",
     journal = {arXiv preprint arXiv:1910.00204},
     archivePrefix = "arXiv",
     eprint = {1910.00204},
     year = 2019,
}

License

Please see the LICENSE file.