{ "source_file": "0930", "source_platform": "Medium", "author": "Riccardo Andreoni", "title_listed": "Advanced Dimensionality Reduction Models Made Simple", "keywords": "", "domain": "机器学习", "content_type": "深度技术解析", "is_paid_flag": "否", "url": "https://towardsdatascience.com/advanced-dimensionality-reduction-models-made-simple-639fca351528/", "publish_date": "2023-11-16", "corpus_stage": "", "notes": "", "id": "17f94bc811be", "collected_at": "2026-10-02T03:17:25", "title": "Advanced Dimensionality Reduction Models Made Simple | Towards Data Science", "body_markdown": "![Revisit consent button](../images/body-df29beacd7496220cc94.svg)\nWe value your privacy\nWe use cookies to enhance your browsing experience, serve personalised ads or content, and analyse our traffic. 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It also helps in adverstising on the website.\n\n\n * Cookie\nedgebucket\n * Duration\nsession\n * Description\nReddit sets this cookie to save the information about a log-on Reddit user, for the purpose of advertisement recommendations and updating the content.\n\n\n * Cookie\ndid\n * Duration\n1 year\n * Description\nArbor sets this cookie to show targeted ads to site visitors.This cookie expires after 2 months or 1 year.\n\n\nUncategorised\nOther uncategorised cookies are those that are being analysed and have not been classified into a category as yet.\nNo cookies to display.\nReject All Save My Preferences Accept All\nToggle Search[](https://towardsdatascience.com/)Toggle Mobile Navigation\n[](https://towardsdatascience.com/)\nPublish AI, ML & data-science insights to a global community of data professionals.\nSign in[Submit an Article](https://write.insightmediagroup.io)\n[Latest](https://towardsdatascience.com/latest/)[Editor's Picks](https://towardsdatascience.com/tag/editors-pick/)[Newsletter](https://towardsdatascience.com/tag/the-variable/)[ShipAI](https://towardsdatascience.com/shipai/)\n* * *\n[Write For TDS](https://towardsdatascience.com/submissions/)\n[LinkedIn](https://www.linkedin.com/company/towards-data-science/)[X](https://x.com/TDataScience)Toggle Search\n[Artificial Intelligence](https://towardsdatascience.com/category/artificial-intelligence/)\n# Advanced Dimensionality Reduction Models Made Simple\nLearn how to efficiently apply state-of-the-art Dimensionality Reduction methods and boost your Machine Learning models.\n[Riccardo Andreoni](https://towardsdatascience.com/author/riccardo.andreoni/)\nNovember 16, 20239 min read\n![Image source: unsplash.com.](../images/000.png)\nImage source: [unsplash.com](https://unsplash.com/photos/multicolored-wall-in-shallow-focus-photography-jbtfM0XBeRc).\nWhen approaching a Machine Learning task, have you ever felt stunned by the **massive number of features**?\nMost Data Scientists experience this overwhelming challenge on a daily basis. While adding features enriches data, it often **slows the training process** and makes it harder to **detect hidden patterns** , resulting in the (in)famous [**Curse of Dimensionality**](https://en.wikipedia.org/wiki/Curse_of_dimensionality)**.**\nMoreover, in high-dimensional spaces **surprising phenomena** happen. To depict this concept with an analogy, think of the novel Flatland, where characters living in a flat (2-dimensional) world find themselves stunned when they encounter a 3-dimensional being. In the same way, we struggle to comprehend that, in high-dimensional spaces, **most of the points are outliers** , and **distances between points are** usually **larger** than expected. All these phenomena, if not treated correctly, may have **disastrous implications** for our Machine Learning models.\n![Image by the author.](../images/001.png)\nImage by the author.\nIn this post, I will explain some advanced Dimensionality Reduction techniques used to mitigate this issue.\nLearn this step by step with the interactive [AI Engineer](https://roadmap.sh/ai-engineer), [AI and Data Scientist](https://roadmap.sh/ai-data-scientist), [Computer Science](https://roadmap.sh/computer-science) and [Machine Learning](https://roadmap.sh/machine-learning) roadmaps.\nIn my previous post, I introduced the relevance of **Dimensionality Reduction** in Machine Learning problems, and how to tame the **Curse of Dimensionality** , and I explained both the theory and Scikit-Learn implementation of the **Principal Component Analysis** algorithm.\nIn this follow-up, I will dive into additional **Dimensionality Reduction algorithms,** like [kPCA](http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.KernelPCA.html) or [LLE](http://scikit-learn.org/stable/modules/generated/sklearn.manifold.LocallyLinearEmbedding.html), that overcome the limitations of PCA.\nDo not worry if you haven't read my Dimensionality Reduction introduction yet. This post is a **stand-alone guide** as I will detail each concept in simple terms. However, if you prefer to know more about PCA, I'm positive this guide will serve your goal:\n> [**Dimensionality Reduction with Scikit-Learn: PCA Theory and Implementation**](https://towardsdatascience.com/dimensionality-reduction-with-scikit-learn-pca-theory-and-implementation-aa224e6ee1f6)\n## Curse of Dimensionality\nThe [Curse of Dimensionality](https://en.wikipedia.org/wiki/Curse_of_dimensionality) is for sure a **significant challenge** in Data Science, making algorithms computationally intensive. When dimensions in a dataset increase, data points become sparse, leading to non-trivial computational issues.\nHumans struggle to conceptualize beyond 3 (or sometimes 2) dimensions, causing **unexpected behavior in high-dimensional spaces**.\nConsider a 2D square: a random point inside it is unlikely to be near a border. However, in higher-dimensional spaces, the probability of a random point to be close to a border grows exponentially. In a 10,000-dimensional hypercube, the probability near a border exceeds 99.99999%. An experiment shows that the average distance between two random points in a 100,000-dimensional hypercube is 129.09, highlighting the vastness of high-dimensional spaces. I will leave you with the code to test these statements yourself.\n> [**articles/dimensionality-reduction/curse-of-dimensionality.ipynb at main · andreoniriccardo/articles**](https://github.com/andreoniriccardo/articles/blob/main/dimensionality-reduction/curse-of-dimensionality.ipynb)\nFor Data Science practitioners, this implies that high-dimensional datasets are **incredibly sparse** , with training instances far apart, raising the risk of overfitting Machine Learning models.\nAdding more training examples isn't practical due to the **exponential increase** in required points as features grow: To maintain a distance of 0.1 between each training point in a 100-feature dataset, you would need to add more points than the number of atoms in the universe. The only viable solution is Dimensionality Reduction, and I will explore techniques to achieve this without significant information loss.\n![Image by the author.](../images/002.png)\nImage by the author.\nFor a broader introduction to the Curse of Dimensionality topic, I link [this article](https://towardsdatascience.com/dimensionality-reduction-with-scikit-learn-pca-theory-and-implementation-aa224e6ee1f6).\nNow it's time to present some of the **most advanced Dimensionality Reduction algorithms**!\n## Kernel PCA\nIndeed the most diffused Dimensionality Reduction model, [Principal Component Analysis](https://en.wikipedia.org/wiki/Principal_component_analysis) (PCA), **projects the data onto a lower-dimensional hyperplane** , lowering the number of dimensions.\nThis simple concept struggles when dealing with datasets that can't be effectively projected on a plane without losing a considerable portion of the original information.\nTake the synthetic Swiss-roll dataset as an example:\n![Image by the author.](../images/003.png)\nImage by the author.\nProjecting data onto a plane is meaningless: **the non-linearity component of the original data is lost**.\nThe most straightforward solution for applying the PCA algorithm to a vast variety of dataset is to exploit the [**Kernel Trick**](https://en.wikipedia.org/wiki/Kernel_method#:~:text=The%20kernel%20trick%20avoids%20the,inner%20product%20in%20another%20space%20.). I presented the Kernel Trick in detail in this post:\n> [**Support Vector Machine with Scikit-Learn: A Friendly Introduction**](https://towardsdatascience.com/support-vector-machine-with-scikit-learn-a-friendly-introduction-a2969f2ff00d)\nThe Kernel Trick maps the data points of a lower-dimensional space to a higher-dimensional one. Support Vector Machine exploits the Kernel Trick to draw linear decision boundaries in high-dimensional spaces, resulting in **complex non-linear boundaries** in the original feature space.\nThe same technique can be applied to the PCA algorithm, resulting in the [**Kernel PCA**](https://en.wikipedia.org/wiki/Kernel_principal_component_analysis) (kPCA).\nApplying the Kernel PCA in Python is easy thanks to the Scikit-learn (sklearn) library. In its simplest form, we just need to instantiate a `KernelPCA()` object, specify the type of kernel to use, and the output's number of dimensions.\npython\n\n```\n# Import required libraries\nfrom sklearn.datasets import make_swiss_roll\nfrom sklearn.decomposition import KernelPCA\n\n# Generate synthetic data\nX_swiss, t = make_swiss_roll(n_samples=1500, noise=0.3, random_state=2)\n\n# Instantiate a KernelPCA object, specifying the kernel type \n# and the output's dimensions\npca_swiss = KernelPCA(n_components=2, kernel='sigmoid', gamma=1e-3, coef0=1, fit_inverse_transform=True)\n# Transform the original data\nX_pca_swiss = pca_swiss.fit_transform(X_swiss)\n\n# Plot the original vs. reduced data\n```\n\nAs a result, we obtain the following reduced dataset:\n![Image by the author.](../images/004.png)\nImage by the author.\nBased on the type of Kernel we pick, kPCA outputs completely different results:\n![Image by the author.](../images/005.png)\nImage by the author.\nIn this simple example, I experimented with 4 different types of Kernels: Linear, Sigmoid, Polynomial, and Cosine. We can immediately appreciate how the Linear and Sigmoid Kernels produce acceptable results with the provided dataset. The Polynomial and Cosine Kernels appear to be poorly suited for this dataset.\nAt this point, you might be asking yourself the following question:\n> Is there a robust method for choosing the right Kernel?\nThe next section aims to solve this issue!\n### How to Select the Kernel\nBeing Kernel PCA an **unsupervised learning model** , we can't simply employ the Grid Search technique and pick the best hyperparameters based on a performance measure. As a consequence selecting the type of Kernel to use is tricky.\nIf kPCA is simply a preprocessing step of your Machine Learning algorithm, followed by a classification or regression model, we can indeed opt for a Grid Search approach. The idea is to treat the type of Kernel of kPCA as one of the hyperparameters of the Machine Learning model and pick the one that yields the best performance measure.\nIf we are treating the kPCA algorithm as a stand-alone model, an alternative consists of choosing the Kernel that yields the lowest reconstruction error.\nTo do that with Scikit-learn, we can exploit the `inverse_transform()` method:\npython\n\n```\nfrom sklearn.datasets import make_swiss_roll\nfrom sklearn.metrics import mean_squared_error\nfrom sklearn.decomposition import KernelPCA\n\nX_swiss, t = make_swiss_roll(n_samples=1500, noise=0.3, random_state=2)\n\nkPCA_dict = {\n \"linear\": KernelPCA(n_components=2, kernel='linear', fit_inverse_transform=True),\n \"sigmoid\": KernelPCA(n_components=2, kernel='sigmoid', gamma=1e-3, coef0=1, fit_inverse_transform=True)\n}\n\nX_pca_swiss = {}\nX_preimage = {}\n\nfor kernel_type in kPCA_dict.keys():\n X_pca_swiss[kernel_type] = kPCA_dict[kernel_type].fit_transform(X_swiss)\n X_preimage[kernel_type] = kPCA_dict[kernel_type].inverse_transform(X_pca_swiss[kernel_type])\n mse = mean_squared_error(X_swiss, X_preimage[kernel_type])\n print('{} kernel -> MSE: {}'.format(kernel_type, mse))\n```\n\n![](../images/body-92f84a011d6b927e43f2.png)\nWe can see how the kPCA with the Sigmoid kernel performs slightly better than the one with the Linear kernel.\n## LLE\nCompletely different from what we saw up to this point, there exists a powerful Dimensionality Reduction technique that doesn't rely on projections. [**Locally Linear Embedding**](https://cs.nyu.edu/~roweis/lle/papers/lleintro.pdf) (LLE) focuses on **preserving local relationships** within the data.\nIn more simple terms, imagine you have a bunch of datapoints in a high-dimensional space, where each dimension represents a different feature. LLE focuses on small groups of neighbouring points, rather than looking at the entire dataset at once. It tries to represent these nearby points in a lower-dimensional space, while maintaining their relationships.\nThe consequent question you might ask now is:\n> What does it mean to maintain the relationships between datapoints?\nWithout involving complex math, for LLE to preserve relationships between datapoints means **maintaining the intrinsic geometric structure of the data** in a lower-dimensional space. Imagine you have a high-dimensional dataset, where certain points are close to each other and share some kind of meaningful relationship. The relationship may depend on several factors like proximity, similarity, connectivity, etc. What it matters is that LLE aims to keep these relationships in place when reducing the dimensionality of the data.\nAs a result, LLE captures the **underlying structure of the data** , especially when the relationships between points are complex and (or) non-linear.\nIn order to compare the LLE algorithm with the kPCA, I will apply LLE to the same dataset and plot the results:\npython\n\n```\nfrom sklearn.datasets import make_swiss_roll\nfrom sklearn.manifold import LocallyLinearEmbedding\nfrom sklearn.decomposition import KernelPCA\n\nX_swiss, t = make_swiss_roll(n_samples=1500, noise=0.3, random_state=2)\n\nlle = LocallyLinearEmbedding(n_components=2, n_neighbors=15)\nX_pca_swiss_lle = lle.fit_transform(X_swiss)\n\npca_swiss = KernelPCA(n_components=2, kernel='sigmoid', gamma=1e-3, coef0=1, fit_inverse_transform=True)\nX_pca_swiss_pca = pca_swiss.fit_transform(X_swiss)\n```\n\n![Image by the author.](../images/006.png)\nImage by the author.\nDifferently from kPCA, LLE appears to \"understand\" better the patterns and relationships of the data and is able to **unroll the Swiss Roll dataset**.\n## PCA vs LLE\nI want to conclude this post about advanced Dimensionality Reduction techniques by delineating the differences between the most commonly used algorithms, and when it is preferable to apply one instead of the other.\nPCA and LLE first diverge on the **approaches** they use. While LLE scans the dataset under a local perspective, focusing on preserving the relationships only between \"adjacent\" points, PCA takes a global viewpoint, capturing the overall dataset variance.\nThe **range of application** of these two different algorithms also differs. By focusing on a small cluster of data, LLE is able to capture complex and non-linear patterns. Instead, PCA, even with the kPCA variation, may fall short at capturing more intricate patterns like the Swiss Roll example we saw before.\nFinally, the way these algorithms are affected by **noise** is a further aspect of difference. PCA, by considering the entire datapoints, tends to be more heavily impacted by noisy datasets, potentially leading to accuracy losses.\n![Image by the author.](../images/007.png)\nImage by the author.\nThese distinct features of LLE and PCA make them suited for **different sets of tasks** and **types of datasets**. LLE is excellent at capturing intricate local structures. Instead, PCA is preferred when the global representation of the dataset and the preservation of the overall variance is a crucial aspect.\nFinally, as the topic of Dimensionality Reduction goes way beyond what a simple Medium post achieves to explain, I recommend diving deeper into this fundamental Machine Learning argument by reading the material I provide in the \"Reference\" section.\n* * *\nIf you liked this story, consider following me to be notified of my upcoming projects and articles!\nHere are some of my past projects:\n> [**Outlier Detection with Scikit-Learn and Matplotlib: a Practical Guide**](https://towardsdatascience.com/outlier-detection-with-scikit-learn-and-matplotlib-a-practical-guide-382d1411b8ec)\n> [**Social Network Analysis with NetworkX: A Gentle Introduction**](https://towardsdatascience.com/social-network-analysis-with-networkx-a-gentle-introduction-6123eddced3)\n> [**Use Deep Learning to Generate Fantasy Names: Build a Language Model from Scratch**](https://towardsdatascience.com/use-deep-learning-to-generate-fantasy-character-names-build-a-language-model-from-scratch-792b13629efa)\n## References\n * [Principal component analysis - Sklearn](http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html)\n * [Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow, 2nd Edition - Aurélien Géron](https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/)\n * [Dive into Deep Learning - Aston Zhang, Zachary C. Lipton, Mu Li, and Alexander J. Smola](https://d2l.ai/)\n * [Schölkopf, B., Smola, A., & Müller, K. (1998). Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10(5), 1299–1319](https://www.mlpack.org/papers/kpca.pdf).\n * [Schölkopf, B., & Smola, A. J. (2002). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press.](https://direct.mit.edu/books/book/1821/Learning-with-KernelsSupport-Vector-Machines)\n * [Mika, S., Schölkopf, B., Smola, A., Müller, K. R., Scholz, M., & Rätsch, G. (1999). Kernel PCA and de-noising in feature spaces. Advances in Neural Information Processing Systems, 11.](https://papers.nips.cc/paper/1491-kernel-pca-and-de-noising-in-feature-spaces)\n * [Roweis, S. T., & Saul, L. K. (2000). Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500), 2323–2326.](https://www.science.org/doi/10.1126/science.290.5500.2323)\n * [Roweis, S. (2002). EM algorithms for PCA and SPCA. Advances in Neural Information Processing Systems, 15.](https://proceedings.neurips.cc/paper/1398-em-algorithms-for-pca-and-spca.pdf)\n * [Saul, L. K., & Roweis, S. T. (2003). Think globally, fit locally: Unsupervised learning of low dimensional manifolds. Journal of Machine Learning Research, 4(Nov), 119–155.](https://psycnet.apa.org/record/2004-13930-001)\n\n\n## Register Now to Continue Reading\n### Unlock Towards Data Science with a Free Account\nAll articles on Towards Data Science are available to registered users. 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It also helps in adverstising on the website.\n\n\n * Cookie\nedgebucket\n * Duration\nsession\n * Description\nReddit sets this cookie to save the information about a log-on Reddit user, for the purpose of advertisement recommendations and updating the content.\n\n\n * Cookie\ndid\n * Duration\n1 year\n * Description\nArbor sets this cookie to show targeted ads to site visitors.This cookie expires after 2 months or 1 year.\n\n\nUncategorised\nOther uncategorised cookies are those that are being analysed and have not been classified into a category as yet.\nNo cookies to display.\nReject All Save My Preferences Accept All\nToggle Search[](https://towardsdatascience.com/)Toggle Mobile Navigation\n[](https://towardsdatascience.com/)\nPublish AI, ML & data-science insights to a global community of data professionals.\nSign in[Submit an Article](https://write.insightmediagroup.io)\n[Latest](https://towardsdatascience.com/latest/)[Editor's Picks](https://towardsdatascience.com/tag/editors-pick/)[Newsletter](https://towardsdatascience.com/tag/the-variable/)[ShipAI](https://towardsdatascience.com/shipai/)\n* * *\n[Write For TDS](https://towardsdatascience.com/submissions/)\n[LinkedIn](https://www.linkedin.com/company/towards-data-science/)[X](https://x.com/TDataScience)Toggle Search\n[Artificial Intelligence](https://towardsdatascience.com/category/artificial-intelligence/)\n# Advanced Dimensionality Reduction Models Made Simple\nLearn how to efficiently apply state-of-the-art Dimensionality Reduction methods and boost your Machine Learning models.\n[Riccardo Andreoni](https://towardsdatascience.com/author/riccardo.andreoni/)\nNovember 16, 20239 min read\n![Image source: unsplash.com.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/1M6hcQOs5qwHhwEBZlBxqKA.png)\nImage source: [unsplash.com](https://unsplash.com/photos/multicolored-wall-in-shallow-focus-photography-jbtfM0XBeRc).\nWhen approaching a Machine Learning task, have you ever felt stunned by the **massive number of features**?\nMost Data Scientists experience this overwhelming challenge on a daily basis. While adding features enriches data, it often **slows the training process** and makes it harder to **detect hidden patterns** , resulting in the (in)famous [**Curse of Dimensionality**](https://en.wikipedia.org/wiki/Curse_of_dimensionality)**.**\nMoreover, in high-dimensional spaces **surprising phenomena** happen. To depict this concept with an analogy, think of the novel Flatland, where characters living in a flat (2-dimensional) world find themselves stunned when they encounter a 3-dimensional being. In the same way, we struggle to comprehend that, in high-dimensional spaces, **most of the points are outliers** , and **distances between points are** usually **larger** than expected. All these phenomena, if not treated correctly, may have **disastrous implications** for our Machine Learning models.\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/148Mp6kwgo4_K6w8FpEDIfQ.png)\nImage by the author.\nIn this post, I will explain some advanced Dimensionality Reduction techniques used to mitigate this issue.\nLearn this step by step with the interactive [AI Engineer](https://roadmap.sh/ai-engineer), [AI and Data Scientist](https://roadmap.sh/ai-data-scientist), [Computer Science](https://roadmap.sh/computer-science) and [Machine Learning](https://roadmap.sh/machine-learning) roadmaps.\nIn my previous post, I introduced the relevance of **Dimensionality Reduction** in Machine Learning problems, and how to tame the **Curse of Dimensionality** , and I explained both the theory and Scikit-Learn implementation of the **Principal Component Analysis** algorithm.\nIn this follow-up, I will dive into additional **Dimensionality Reduction algorithms,** like [kPCA](http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.KernelPCA.html) or [LLE](http://scikit-learn.org/stable/modules/generated/sklearn.manifold.LocallyLinearEmbedding.html), that overcome the limitations of PCA.\nDo not worry if you haven't read my Dimensionality Reduction introduction yet. This post is a **stand-alone guide** as I will detail each concept in simple terms. However, if you prefer to know more about PCA, I'm positive this guide will serve your goal:\n> [**Dimensionality Reduction with Scikit-Learn: PCA Theory and Implementation**](https://towardsdatascience.com/dimensionality-reduction-with-scikit-learn-pca-theory-and-implementation-aa224e6ee1f6)\n## Curse of Dimensionality\nThe [Curse of Dimensionality](https://en.wikipedia.org/wiki/Curse_of_dimensionality) is for sure a **significant challenge** in Data Science, making algorithms computationally intensive. When dimensions in a dataset increase, data points become sparse, leading to non-trivial computational issues.\nHumans struggle to conceptualize beyond 3 (or sometimes 2) dimensions, causing **unexpected behavior in high-dimensional spaces**.\nConsider a 2D square: a random point inside it is unlikely to be near a border. However, in higher-dimensional spaces, the probability of a random point to be close to a border grows exponentially. In a 10,000-dimensional hypercube, the probability near a border exceeds 99.99999%. An experiment shows that the average distance between two random points in a 100,000-dimensional hypercube is 129.09, highlighting the vastness of high-dimensional spaces. I will leave you with the code to test these statements yourself.\n> [**articles/dimensionality-reduction/curse-of-dimensionality.ipynb at main · andreoniriccardo/articles**](https://github.com/andreoniriccardo/articles/blob/main/dimensionality-reduction/curse-of-dimensionality.ipynb)\nFor Data Science practitioners, this implies that high-dimensional datasets are **incredibly sparse** , with training instances far apart, raising the risk of overfitting Machine Learning models.\nAdding more training examples isn't practical due to the **exponential increase** in required points as features grow: To maintain a distance of 0.1 between each training point in a 100-feature dataset, you would need to add more points than the number of atoms in the universe. The only viable solution is Dimensionality Reduction, and I will explore techniques to achieve this without significant information loss.\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/12A2bwguQkTRWvCBwAGu7gw.png)\nImage by the author.\nFor a broader introduction to the Curse of Dimensionality topic, I link [this article](https://towardsdatascience.com/dimensionality-reduction-with-scikit-learn-pca-theory-and-implementation-aa224e6ee1f6).\nNow it's time to present some of the **most advanced Dimensionality Reduction algorithms**!\n## Kernel PCA\nIndeed the most diffused Dimensionality Reduction model, [Principal Component Analysis](https://en.wikipedia.org/wiki/Principal_component_analysis) (PCA), **projects the data onto a lower-dimensional hyperplane** , lowering the number of dimensions.\nThis simple concept struggles when dealing with datasets that can't be effectively projected on a plane without losing a considerable portion of the original information.\nTake the synthetic Swiss-roll dataset as an example:\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/1QzBZJCE1Dcn6lPyVzko3Hg.png)\nImage by the author.\nProjecting data onto a plane is meaningless: **the non-linearity component of the original data is lost**.\nThe most straightforward solution for applying the PCA algorithm to a vast variety of dataset is to exploit the [**Kernel Trick**](https://en.wikipedia.org/wiki/Kernel_method#:~:text=The%20kernel%20trick%20avoids%20the,inner%20product%20in%20another%20space%20.). I presented the Kernel Trick in detail in this post:\n> [**Support Vector Machine with Scikit-Learn: A Friendly Introduction**](https://towardsdatascience.com/support-vector-machine-with-scikit-learn-a-friendly-introduction-a2969f2ff00d)\nThe Kernel Trick maps the data points of a lower-dimensional space to a higher-dimensional one. Support Vector Machine exploits the Kernel Trick to draw linear decision boundaries in high-dimensional spaces, resulting in **complex non-linear boundaries** in the original feature space.\nThe same technique can be applied to the PCA algorithm, resulting in the [**Kernel PCA**](https://en.wikipedia.org/wiki/Kernel_principal_component_analysis) (kPCA).\nApplying the Kernel PCA in Python is easy thanks to the Scikit-learn (sklearn) library. In its simplest form, we just need to instantiate a `KernelPCA()` object, specify the type of kernel to use, and the output's number of dimensions.\npython\n\n```\n# Import required libraries\nfrom sklearn.datasets import make_swiss_roll\nfrom sklearn.decomposition import KernelPCA\n\n# Generate synthetic data\nX_swiss, t = make_swiss_roll(n_samples=1500, noise=0.3, random_state=2)\n\n# Instantiate a KernelPCA object, specifying the kernel type \n# and the output's dimensions\npca_swiss = KernelPCA(n_components=2, kernel='sigmoid', gamma=1e-3, coef0=1, fit_inverse_transform=True)\n# Transform the original data\nX_pca_swiss = pca_swiss.fit_transform(X_swiss)\n\n# Plot the original vs. reduced data\n```\n\nAs a result, we obtain the following reduced dataset:\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/19d7_X7VVaFULdxaJI-41tA.png)\nImage by the author.\nBased on the type of Kernel we pick, kPCA outputs completely different results:\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/1gzU05WqJm-uioCSZ8Nf1Kw.png)\nImage by the author.\nIn this simple example, I experimented with 4 different types of Kernels: Linear, Sigmoid, Polynomial, and Cosine. We can immediately appreciate how the Linear and Sigmoid Kernels produce acceptable results with the provided dataset. The Polynomial and Cosine Kernels appear to be poorly suited for this dataset.\nAt this point, you might be asking yourself the following question:\n> Is there a robust method for choosing the right Kernel?\nThe next section aims to solve this issue!\n### How to Select the Kernel\nBeing Kernel PCA an **unsupervised learning model** , we can't simply employ the Grid Search technique and pick the best hyperparameters based on a performance measure. As a consequence selecting the type of Kernel to use is tricky.\nIf kPCA is simply a preprocessing step of your Machine Learning algorithm, followed by a classification or regression model, we can indeed opt for a Grid Search approach. The idea is to treat the type of Kernel of kPCA as one of the hyperparameters of the Machine Learning model and pick the one that yields the best performance measure.\nIf we are treating the kPCA algorithm as a stand-alone model, an alternative consists of choosing the Kernel that yields the lowest reconstruction error.\nTo do that with Scikit-learn, we can exploit the `inverse_transform()` method:\npython\n\n```\nfrom sklearn.datasets import make_swiss_roll\nfrom sklearn.metrics import mean_squared_error\nfrom sklearn.decomposition import KernelPCA\n\nX_swiss, t = make_swiss_roll(n_samples=1500, noise=0.3, random_state=2)\n\nkPCA_dict = {\n \"linear\": KernelPCA(n_components=2, kernel='linear', fit_inverse_transform=True),\n \"sigmoid\": KernelPCA(n_components=2, kernel='sigmoid', gamma=1e-3, coef0=1, fit_inverse_transform=True)\n}\n\nX_pca_swiss = {}\nX_preimage = {}\n\nfor kernel_type in kPCA_dict.keys():\n X_pca_swiss[kernel_type] = kPCA_dict[kernel_type].fit_transform(X_swiss)\n X_preimage[kernel_type] = kPCA_dict[kernel_type].inverse_transform(X_pca_swiss[kernel_type])\n mse = mean_squared_error(X_swiss, X_preimage[kernel_type])\n print('{} kernel -> MSE: {}'.format(kernel_type, mse))\n```\n\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/1q7oJ7teMWsgC0sr3WCtIBA.png)\nWe can see how the kPCA with the Sigmoid kernel performs slightly better than the one with the Linear kernel.\n## LLE\nCompletely different from what we saw up to this point, there exists a powerful Dimensionality Reduction technique that doesn't rely on projections. [**Locally Linear Embedding**](https://cs.nyu.edu/~roweis/lle/papers/lleintro.pdf) (LLE) focuses on **preserving local relationships** within the data.\nIn more simple terms, imagine you have a bunch of datapoints in a high-dimensional space, where each dimension represents a different feature. LLE focuses on small groups of neighbouring points, rather than looking at the entire dataset at once. It tries to represent these nearby points in a lower-dimensional space, while maintaining their relationships.\nThe consequent question you might ask now is:\n> What does it mean to maintain the relationships between datapoints?\nWithout involving complex math, for LLE to preserve relationships between datapoints means **maintaining the intrinsic geometric structure of the data** in a lower-dimensional space. Imagine you have a high-dimensional dataset, where certain points are close to each other and share some kind of meaningful relationship. The relationship may depend on several factors like proximity, similarity, connectivity, etc. What it matters is that LLE aims to keep these relationships in place when reducing the dimensionality of the data.\nAs a result, LLE captures the **underlying structure of the data** , especially when the relationships between points are complex and (or) non-linear.\nIn order to compare the LLE algorithm with the kPCA, I will apply LLE to the same dataset and plot the results:\npython\n\n```\nfrom sklearn.datasets import make_swiss_roll\nfrom sklearn.manifold import LocallyLinearEmbedding\nfrom sklearn.decomposition import KernelPCA\n\nX_swiss, t = make_swiss_roll(n_samples=1500, noise=0.3, random_state=2)\n\nlle = LocallyLinearEmbedding(n_components=2, n_neighbors=15)\nX_pca_swiss_lle = lle.fit_transform(X_swiss)\n\npca_swiss = KernelPCA(n_components=2, kernel='sigmoid', gamma=1e-3, coef0=1, fit_inverse_transform=True)\nX_pca_swiss_pca = pca_swiss.fit_transform(X_swiss)\n```\n\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/12OCyqLnZLfZCxVDzMLnL4Q.png)\nImage by the author.\nDifferently from kPCA, LLE appears to \"understand\" better the patterns and relationships of the data and is able to **unroll the Swiss Roll dataset**.\n## PCA vs LLE\nI want to conclude this post about advanced Dimensionality Reduction techniques by delineating the differences between the most commonly used algorithms, and when it is preferable to apply one instead of the other.\nPCA and LLE first diverge on the **approaches** they use. While LLE scans the dataset under a local perspective, focusing on preserving the relationships only between \"adjacent\" points, PCA takes a global viewpoint, capturing the overall dataset variance.\nThe **range of application** of these two different algorithms also differs. By focusing on a small cluster of data, LLE is able to capture complex and non-linear patterns. Instead, PCA, even with the kPCA variation, may fall short at capturing more intricate patterns like the Swiss Roll example we saw before.\nFinally, the way these algorithms are affected by **noise** is a further aspect of difference. PCA, by considering the entire datapoints, tends to be more heavily impacted by noisy datasets, potentially leading to accuracy losses.\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/1KfdsbYJ9AOEhLGK5EZ_aXA.png)\nImage by the author.\nThese distinct features of LLE and PCA make them suited for **different sets of tasks** and **types of datasets**. LLE is excellent at capturing intricate local structures. Instead, PCA is preferred when the global representation of the dataset and the preservation of the overall variance is a crucial aspect.\nFinally, as the topic of Dimensionality Reduction goes way beyond what a simple Medium post achieves to explain, I recommend diving deeper into this fundamental Machine Learning argument by reading the material I provide in the \"Reference\" section.\n* * *\nIf you liked this story, consider following me to be notified of my upcoming projects and articles!\nHere are some of my past projects:\n> [**Outlier Detection with Scikit-Learn and Matplotlib: a Practical Guide**](https://towardsdatascience.com/outlier-detection-with-scikit-learn-and-matplotlib-a-practical-guide-382d1411b8ec)\n> [**Social Network Analysis with NetworkX: A Gentle Introduction**](https://towardsdatascience.com/social-network-analysis-with-networkx-a-gentle-introduction-6123eddced3)\n> [**Use Deep Learning to Generate Fantasy Names: Build a Language Model from Scratch**](https://towardsdatascience.com/use-deep-learning-to-generate-fantasy-character-names-build-a-language-model-from-scratch-792b13629efa)\n## References\n * [Principal component analysis - Sklearn](http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.PCA.html)\n * [Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow, 2nd Edition - Aurélien Géron](https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/)\n * [Dive into Deep Learning - Aston Zhang, Zachary C. Lipton, Mu Li, and Alexander J. Smola](https://d2l.ai/)\n * [Schölkopf, B., Smola, A., & Müller, K. (1998). Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10(5), 1299–1319](https://www.mlpack.org/papers/kpca.pdf).\n * [Schölkopf, B., & Smola, A. J. (2002). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press.](https://direct.mit.edu/books/book/1821/Learning-with-KernelsSupport-Vector-Machines)\n * [Mika, S., Schölkopf, B., Smola, A., Müller, K. R., Scholz, M., & Rätsch, G. (1999). Kernel PCA and de-noising in feature spaces. Advances in Neural Information Processing Systems, 11.](https://papers.nips.cc/paper/1491-kernel-pca-and-de-noising-in-feature-spaces)\n * [Roweis, S. T., & Saul, L. K. (2000). Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500), 2323–2326.](https://www.science.org/doi/10.1126/science.290.5500.2323)\n * [Roweis, S. (2002). EM algorithms for PCA and SPCA. Advances in Neural Information Processing Systems, 15.](https://proceedings.neurips.cc/paper/1398-em-algorithms-for-pca-and-spca.pdf)\n * [Saul, L. K., & Roweis, S. T. (2003). Think globally, fit locally: Unsupervised learning of low dimensional manifolds. Journal of Machine Learning Research, 4(Nov), 119–155.](https://psycnet.apa.org/record/2004-13930-001)\n\n\n## Register Now to Continue Reading\n### Unlock Towards Data Science with a Free Account\nAll articles on Towards Data Science are available to registered users. 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