Contrastive Neighbor Embeddings

Parametric and nonparametric neighbor embeddings suitable for data visualization with various contrastive losses.

Reference:

• From t-SNE to UMAP with contrastive learning, ICLR 2023
  Sebastian Damrich, Niklas Böhm, Fred A Hamprecht, Dmitry Kobak

@inproceedings{damrich2023from, title={From $t$-{SNE} to {UMAP} with contrastive learning}, author={Damrich, Sebastian and B{\"o}hm, Jan Niklas and Hamprecht, Fred A and Kobak, Dmitry}, booktitle={International Conference on Learning Representations}, year={2023}, }

This repository provides our PyTorch library. The code that implements the specific analysis presented in the paper is available at https://github.com/hci-unihd/cl-tsne-umap.

Scope

This repository allows to use several different losses, training modes, devices, and distance measures. It (re-)implements the UMAP loss[^umap], the negative sampling loss (NEG)[^neg], noise-contrastive estimation loss (NCE)[^nce], and InfoNCE loss[^infonce] in PyTorch. All of these losses can be combined with embedding similarities either based on the Cauchy distribution (default) or on the cosine distance. The embedding positions can either be optimized directly (non-parametric mode) or a neural network can be trained to map data to embedding positions (parametric mode). Our pure PyTorch implementation can run seamlessly on CPU or GPU.

As a result, this library re-implements several existing contrastive methods, alongside many new ones. The most important ones are summarized the table below.

| Loss | Non-parametric | Parametric | |-------------------|-------------------|--------------------------------| | UMAP[^umap] | UMAP[^umap] | Parametric UMAP[^pumap] | | NEG[^neg] | Neg-t-SNE (new) | Parametric Neg-t-SNE (new) | | NCE[^nce] | NCVis[^ncvis] | Parametric NCVis (new) | | InfoNCE[^infonce] | InfoNC-t-SNE (new) | Parametric InfoNC-t-SNE (new) |

The repository can also be used to run SimCLR[^simclr] experiments, by using the InfoNCE loss. The main class ContrastiveEmbedding allows to change the similarity measure to the exponential of a temperature-scaled cosine similarity (metric="cosine"). Its forward method accepts a dataloader. If the dataloader implements data augmentation, one obtains SimCLR.

Installation

Pip installation: sh pip install contrastive-ne

To install from source, clone this repository sh git clone https://github.com/berenslab/contrastive-ne cd contrastive-ne pip install .

This installs all dependecies and allows the code to be run. Note that pytorch with GPU support can be a bit tricky to install as a dependency, so if it is not installed already, it might make sense to consult the pytorch website to install it with CUDA support prior to the installation of contrastive-ne.

Example

The most basic usage is via the CNE class. Here are some Hello World examples using the MNIST dataset.

```python import cne import torchvision import numpy as np import matplotlib.pyplot as plt

load MNIST

mnisttrain = torchvision.datasets.MNIST(train=True, download=True, transform=None) xtrain, ytrain = mnisttrain.data.float().numpy(), mnist_train.targets

mnisttest = torchvision.datasets.MNIST(train=False, download=True, transform=None) xtest, ytest = mnisttest.data.float().numpy(), mnist_test.targets

xtrain = xtrain.reshape(xtrain.shape[0], -1) xtest = xtest.reshape(xtest.shape[0], -1)

x = np.concatenate([xtrain, xtest], axis=0) y = np.concatenate([ytrain, ytest], axis=0) ```

By default, CNE uses the InfoNCE loss and thus approximates $t$-SNE (one can increase the number of negative samples, e.g. negative_samples=500 to get a better approximation, see also below):

```python

default CNE (using the InfoNCE loss)

embedder = cne.CNE() embd = embedder.fit_transform(x)

plt.figure() plt.scatter(*embd.T, c=y, alpha=0.5, s=1.0, cmap="tab10", edgecolor="none") plt.gca().set_aspect("equal") plt.axis("off") plt.title(r"InfoNC-$t$-SNE of MNIST") plt.show() ```

To get non-parametric Neg-t-SNE (very close to UMAP) use loss_mode="neg":

```python

non-parametric Neg-t-SNE

embedderneg = cne.CNE(lossmode="neg") embdneg = embedderneg.fit_transform(x)

plt.figure() plt.scatter(*embdneg.T, c=y, alpha=0.5, s=1.0, cmap="tab10", edgecolor="none") plt.gca().setaspect("equal") plt.axis("off") plt.title(r"Neg-$t$-SNE of MNIST") plt.show() ```

Here is a parametric NCVis (NC-t-SNE) example, highlighting that new embedding points can be added with a parametric embedding:

```python

parametric NCVis, highlighting the embedding of new points

embedderncvis = cne.CNE(lossmode="nce", optimizer="adam", # Adam tends to work better than SGD for parametric runs parametric=True) embdncvistrain = embedderncvis.fittransform(xtrain) # only train with training set embdncvistest = embedderncvis.transform(x_test) # transform test set with the trained model

plot

titles = ["Train", "Test"] fig, ax = plt.subplots(1, 2, figsize=(5.5, 2.5), constrainedlayout=True) ax[0].scatter(*embdncvistrain.T, c=ytrain, alpha=0.5, s=1.0, cmap="tab10", edgecolor="none") ax[1].scatter(*embdncvistest.T, c=y_test, alpha=0.5, s=1.0, cmap="tab10", edgecolor="none")

for i in range(2): ax[i].settitle(titles[i]) ax[i].setaspect("equal", "datalim") ax[i].axis("off")

fig.suptitle("Parametric NCVis of MNIST") plt.show() ```

To compute the spectrum of neighbor embeddings with the negative sampling loss, we can use the following code:

```python

compute spectrum with negative sampling loss

spec_params = [0.0, 0.5, 1.0]

negembeddings = {} for s in specparams: embedder = cne.CNE(lossmode="neg", s=s) embd = embedder.fittransform(x) neg_embeddings[s] = embd

plot embeddings

fig, ax = plt.subplots(1, len(specparams), figsize=(5.5, 3), constrainedlayout=True) for i, s in enumerate(specparams): ax[i].scatter(*negembeddings[s].T, c=y, alpha=0.5, s=0.1, cmap="tab10", edgecolor="none") ax[i].setaspect("equal", "datalim") ax[i].axis("off") ax[i].settitle(f"s={s}")

fig.suptitle("Negative sampling spectrum of MNIST") plt.show() ```

A similar spectrum can be computed using the InfoNCE loss (note that this is not described in our paper but was implemented after it has been published). Using a higher number of negative samples leads to better local structure: ```python

compute spectrum with InfoNCE loss

spec_params = [0.0, 0.5, 1.0]

inceembeddings = {} for s in specparams: embedder = cne.CNE(negativesamples=500, # more negative samples for better local quality s=s) embd = embedder.fittransform(x) ince_embeddings[s] = embd

plot embeddings

fig, ax = plt.subplots(1, len(specparams), figsize=(5.5, 3), constrainedlayout=True) for i, s in enumerate(specparams): ax[i].scatter(*inceembeddings[s].T, c=y, alpha=0.5, s=0.1, cmap="tab10", edgecolor="none") ax[i].setaspect("equal", "datalim") ax[i].axis("off") ax[i].settitle(f"s={s}")

fig.suptitle("InfoNCE spectrum of MNIST") plt.show() ```

Contrastive neighbor embedding spectra

The CNE class takes an argument s which indicates the position of the embedding on the attraction-repulsion spectrum, where s=0 corresponds to a t-SNE-like embedding and s=1 corresponds to a UMAP-like embedding. The spectrum is implemented for the negative sampling loss mode (loss_mode=neg) and the InfoNCE loss mode (loss_mode=infonce). It implements a trade-off between preserving discrete (local) and continuous (global) structure.

For a more fine-grained control, there are arguments specific to the loss mode. For negative sampling one can specify the argument Z_bar which directly sets the fixed normalization constant or neg_spec which sets the value in the denominator of the negative sampling estimator. Their relation is Z_bar = m * neg_spec / n**2 where n is the sample size and m the number of negative samples. The high-level argument s corresponds to Z_bar = 100 * n for s=0 and Z_bar = n**2 / m for s=1. Note that the s=1 value is exactly what UMAP uses, whereas the s=0 value is our heuristic that usually approximates t-SNE well.

For InfoNCE, the argument ince_spec controls the exaggeration of attraction over repulsion. The setting s=0 corresponds to ince_spec=1 and s=1 corresponds to ince_spec=4.0. Here s=0 recovers t-SNE value exactly, whereas s=1 is our heuristic that usually approximates UMAP well.

For all input arguments controlling the position on the spectrum, larger values yield more attraction and thus better global structure preservation, while smaller values lead to a focus on the local structure.

Early exaggeration

Similar to early exaggeration in t-SNE the loss modes neg and infonce use early exaggeration by default. To disable it pass early_exaggeration=False to the CNE class. If used, the first third of the optimization epochs use the setting s=1.0 unless a higher value s is specified. This is to ensure that the embedding is initialized with a good global structure.

Technical details

The object ContrastiveEmbedding needs a neural network model as a required parameter in order to be created. The fit method then takes a torch.utils.data.DataLoader that will be used for training. The data loader returns a pair of two neighboring points. In the classical NE setting this would be two points that share an edge in the kNN graph; the contrastive self-supervised learning approach will transform a sample twice and return those as a “positive edge” which will denote the attractive force between the two points.

Run time analysis

The run time depends strongly on the training mode (parametric / non-parametric), the device (CPU / GPU) and on the batch size. The plot below illustrates this for the optimization of a Neg-t-SNE embedding of MNIST. Note that the non-parametric setting on GPU becomes competitive with the reference (CPU) implementations of UMAP[^umap] and t-SNE[^tsne].

Logging

There are callbacks for logging the embeddings and losses during training. Note that the loss logging depends on the vis_utils repository, which needs to be installed separately.

[^umap]: McInnes, Leland, John Healy, and James Melville. "UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction." arXiv preprint arXiv:1802.03426 (2018).
[^nce]: Gutmann, Michael U., and Aapo Hyvärinen. "Noise-Contrastive Estimation of Unnormalized Statistical Models, with Applications to Natural Image Statistics." Journal of Machine Learning Research 13.2 (2012).
[^neg]: Mikolov, Tomas, et al. "Distributed Representations of Words and Phrases and their Compositionality." Advances in Neural Information Processing Systems 26 (2013).
[^infonce]: Oord, Aaron van den, Yazhe Li, and Oriol Vinyals. "Representation Learning with Contrastive Predictive Coding." arXiv preprint arXiv:1807.03748 (2018).
[^pumap]: Sainburg, Tim, Leland McInnes, and Timothy Q. Gentner. "Parametric UMAP Embeddings for Representation and Semisupervised Learning." Neural Computation 33.11 (2021): 2881-2907.
[^ncvis]: Artemenkov, Aleksandr, and Maxim Panov. "NCVis: Noise Contrastive Approach for Scalable Visualization." Proceedings of The Web Conference 2020. 2020.
[^simclr]: Chen, Ting, et al. "A Simple Framework for Contrastive Learning of Visual Representations." International conference on machine learning. PMLR, 2020.
[^tsne]: Poličar, Pavlin G., Martin Stražar, and Blaž Zupan. "openTSNE: a modular Python library for t-SNE dimensionality reduction and embedding." BioRxiv (2019): 731877.