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lw2/README.md
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# Labwork 2 - Clustering
## K-means
Select two datasets from UCI, and for each of them:
1. Run experiments with k-means. Explain the experimental protocol.
2. Explain the centroid initialization of the k-means used in the first question.
3. Analyze and compare the results with different values of k.
4. Calculate the clustering quality (any criteria in slides).
## Subspace clustering
1. Select one high-dimensional dataset (>100 features)
2. Use PCA or SVD to visualize the data distribution in 2D/3D.
3. Apply any clustering method (DBSCAN, K-means, SOM, GNG, etc.) for the selected datasets
in 2D/3D after using PCA/SVD. Compare the performance before and after the dimensionality reduction.
4. Select a random subspace of the dataset. Re-apply any clustering method on the subspace.
Discuss and compare the performance with the previous results.
lw2/c.bib
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@@ -5,6 +5,9 @@
howpublished = {UCI Machine Learning Repository},
note = {{DOI}: https://doi.org/10.24432/C53W3X}
}
Redmond, Michael. 2002. Communities and Crime. UCI Machine Learning Repository. https://doi.org/10.24432/C53W3X.
@misc{abalone_1,
author = {Nash, Warwick, Sellers, Tracy, Talbot, Simon, Cawthorn, Andrew, and Ford, Wes},
title = {{Abalone}},
@@ -12,10 +15,15 @@
howpublished = {UCI Machine Learning Repository},
note = {{DOI}: https://doi.org/10.24432/C55C7W}
}
Nash, Warwick, Tracy Sellers, Simon Talbot, Andrew Cawthorn, and Wes Ford. 1994. Abalone. UCI Machine Learning Repository. https://doi.org/10.24432/C55C7W.
@misc{covertype_31,
author = {Blackard, Jock},
title = {{Covertype}},
year = {1998},
howpublished = {UCI Machine Learning Repository},
note = {{DOI}: https://doi.org/10.24432/C50K5N}
}
}
Blackard, Jock. 1998. Covertype. UCI Machine Learning Repository. https://doi.org/10.24432/C50K5N.
lw2/lw2.ipynb
+183 -85
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@@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "code",
"execution_count": 116,
"execution_count": 32,
"id": "e377bc9b",
"metadata": {},
"outputs": [],
@@ -13,14 +13,18 @@
"from sklearn.cluster import KMeans, MiniBatchKMeans\n",
"from sklearn.decomposition import PCA\n",
"from sklearn.preprocessing import StandardScaler\n",
"from sklearn.metrics import silhouette_score\n",
"from sklearn.metrics import (\n",
" silhouette_score,\n",
" davies_bouldin_score,\n",
" calinski_harabasz_score,\n",
")\n",
"\n",
"from pathlib import Path\n"
"from pathlib import Path"
]
},
{
"cell_type": "code",
"execution_count": 117,
"execution_count": 33,
"id": "d7275d15",
"metadata": {},
"outputs": [
@@ -30,7 +34,7 @@
"WindowsPath('c:/Users/Admin/Documents/lecture ass shit/3/ml2/lw2')"
]
},
"execution_count": 117,
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
@@ -51,7 +55,7 @@
},
{
"cell_type": "code",
"execution_count": 118,
"execution_count": 34,
"id": "4de18adb",
"metadata": {},
"outputs": [],
@@ -78,7 +82,7 @@
},
{
"cell_type": "code",
"execution_count": 119,
"execution_count": 35,
"id": "50b3eb52",
"metadata": {},
"outputs": [],
@@ -105,45 +109,28 @@
},
{
"cell_type": "code",
"execution_count": 120,
"execution_count": 36,
"id": "a89a6974",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([0.4307184563069171,\n",
" 0.3308881531841743,\n",
" 0.34454881088271005,\n",
" 0.3330973959780588,\n",
" 0.37350567261284556,\n",
" 0.39200932042425607,\n",
" 0.3905976225423938,\n",
" 0.38566484346143776,\n",
" 0.3910649022379036],\n",
" [18066.009207127645,\n",
" 13418.727667404704,\n",
" 11157.455698492053,\n",
" 9330.808564389003,\n",
" 7660.230080453761,\n",
" 6764.331897239782,\n",
" 5939.9031003732325,\n",
" 5315.323604570619,\n",
" 4805.410264465898])"
"'silhouettes = [0.4307184563069171, 0.3308881531841743, 0.34454881088271005, 0.3330973959780588, 0.37350567261284556, 0.39200932042425607, 0.3905976225423938, 0.38566484346143776, 0.3910649022379036], inertias = [18066.00920712765, 13418.727667404704, 11157.455698492053, 9330.808564389003, 7660.230080453761, 6764.331897239782, 5939.903100373233, 5315.323604570618, 4805.410264465897]'"
]
},
"execution_count": 120,
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"silhouettes, inertias"
"f\"{silhouettes = }, {inertias = }\""
]
},
{
"cell_type": "code",
"execution_count": 121,
"execution_count": 37,
"id": "c70bfa21",
"metadata": {},
"outputs": [],
@@ -154,7 +141,7 @@
},
{
"cell_type": "code",
"execution_count": 122,
"execution_count": 38,
"id": "926a255e",
"metadata": {},
"outputs": [
@@ -179,7 +166,7 @@
},
{
"cell_type": "code",
"execution_count": 123,
"execution_count": 39,
"id": "7b8c1750",
"metadata": {},
"outputs": [
@@ -204,7 +191,7 @@
},
{
"cell_type": "code",
"execution_count": 124,
"execution_count": 40,
"id": "acab8d95",
"metadata": {},
"outputs": [
@@ -227,6 +214,33 @@
"plt.show()\n"
]
},
{
"cell_type": "code",
"execution_count": 41,
"id": "b7447f18",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Abalone: n_samples=4177, n_features=9\n",
"Best k: 2\n",
"Cluster sizes: [1940 2237]\n",
"Davies-Bouldin Index: 0.8836\n",
"Calinski-Harabasz Index: 4512.6320\n"
]
}
],
"source": [
"# Abalone dataset stats\n",
"print(f\"Abalone: n_samples={X.shape[0]}, n_features={X.shape[1]}\")\n",
"print(f\"Best k: {best_k}\")\n",
"print(f\"Cluster sizes: {np.bincount(best_labels)}\")\n",
"print(f\"Davies-Bouldin Index: {davies_bouldin_score(X, best_labels):.4f}\")\n",
"print(f\"Calinski-Harabasz Index: {calinski_harabasz_score(X, best_labels):.4f}\")"
]
},
{
"cell_type": "markdown",
"id": "3841acb2",
@@ -237,7 +251,7 @@
},
{
"cell_type": "code",
"execution_count": 125,
"execution_count": 42,
"id": "92b692bc",
"metadata": {},
"outputs": [],
@@ -252,7 +266,7 @@
},
{
"cell_type": "code",
"execution_count": 126,
"execution_count": 43,
"id": "e6bbfcb7",
"metadata": {},
"outputs": [],
@@ -277,41 +291,28 @@
},
{
"cell_type": "code",
"execution_count": 127,
"execution_count": 44,
"id": "05d4bd31",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([-0.030689560868985374,\n",
" 0.014267829228328097,\n",
" 0.052170441001039715,\n",
" 0.0763160133542805,\n",
" 0.05712772597563029,\n",
" 0.11281790580873496,\n",
" 0.1241172489653034],\n",
" [1033855.8510931745,\n",
" 983031.2462602488,\n",
" 933199.3143504256,\n",
" 914886.5211138967,\n",
" 889334.3440412064,\n",
" 868852.814198654,\n",
" 848656.9204411206])"
"'silhouettes = [-0.030689560868985374, 0.014267829228328097, 0.052170441001039715, 0.0763160133542805, 0.05712772597563029, 0.11281790580873496, 0.1241172489653034], inertias = [1033855.8510931745, 983031.2462602488, 933199.3143504256, 914886.5211138967, 889334.3440412064, 868852.814198654, 848656.9204411206]'"
]
},
"execution_count": 127,
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"silhouettes, inertias"
"f\"{silhouettes = }, {inertias = }\""
]
},
{
"cell_type": "code",
"execution_count": 128,
"execution_count": 45,
"id": "da938557",
"metadata": {},
"outputs": [
@@ -336,7 +337,7 @@
},
{
"cell_type": "code",
"execution_count": 129,
"execution_count": 46,
"id": "0070102e",
"metadata": {},
"outputs": [
@@ -361,7 +362,7 @@
},
{
"cell_type": "code",
"execution_count": 130,
"execution_count": 47,
"id": "c312a24a",
"metadata": {},
"outputs": [],
@@ -370,6 +371,33 @@
"best_labels = labels_per_k[best_k]"
]
},
{
"cell_type": "code",
"execution_count": 48,
"id": "71ae758d",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Covertype: n_samples=20000, n_features=54\n",
"Best k: 8\n",
"Cluster sizes: [1407 6092 3757 2258 1139 4226 88 1033]\n",
"Davies-Bouldin Index: 2.3425\n",
"Calinski-Harabasz Index: 714.6171\n"
]
}
],
"source": [
"# Covertype dataset stats\n",
"print(f\"Covertype: n_samples={X.shape[0]}, n_features={X.shape[1]}\")\n",
"print(f\"Best k: {best_k}\")\n",
"print(f\"Cluster sizes: {np.bincount(best_labels)}\")\n",
"print(f\"Davies-Bouldin Index: {davies_bouldin_score(X, best_labels):.4f}\")\n",
"print(f\"Calinski-Harabasz Index: {calinski_harabasz_score(X, best_labels):.4f}\")"
]
},
{
"cell_type": "markdown",
"id": "aecfb074",
@@ -380,7 +408,7 @@
},
{
"cell_type": "code",
"execution_count": 131,
"execution_count": 49,
"id": "bc26af01",
"metadata": {},
"outputs": [],
@@ -408,7 +436,7 @@
},
{
"cell_type": "code",
"execution_count": 132,
"execution_count": 50,
"id": "a3e10a92",
"metadata": {},
"outputs": [
@@ -434,6 +462,32 @@
"plt.show()\n"
]
},
{
"cell_type": "code",
"execution_count": 51,
"id": "1a38e68b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Communities: n_samples=1994, n_features=122\n",
"Components for 90% variance: 30\n",
"First 2 components explain: 35.0%\n"
]
}
],
"source": [
"# Communities dataset stats + PCA 90% variance\n",
"print(f\"Communities: n_samples={X.shape[0]}, n_features={X.shape[1]}\")\n",
"n_90 = np.argmax(np.cumsum(pca.explained_variance_ratio_) >= 0.90) + 1\n",
"print(f\"Components for 90% variance: {n_90}\")\n",
"print(\n",
" f\"First 2 components explain: {pca.explained_variance_ratio_[:2].sum() * 100:.1f}%\"\n",
")"
]
},
{
"cell_type": "markdown",
"id": "a7acc3d1",
@@ -444,7 +498,7 @@
},
{
"cell_type": "code",
"execution_count": 133,
"execution_count": 52,
"id": "69a1249a",
"metadata": {},
"outputs": [
@@ -477,7 +531,7 @@
},
{
"cell_type": "code",
"execution_count": 134,
"execution_count": 53,
"id": "f41491b4",
"metadata": {},
"outputs": [],
@@ -501,7 +555,7 @@
},
{
"cell_type": "code",
"execution_count": 135,
"execution_count": 54,
"id": "873caac5",
"metadata": {},
"outputs": [],
@@ -526,7 +580,7 @@
},
{
"cell_type": "code",
"execution_count": 136,
"execution_count": 55,
"id": "6846b929",
"metadata": {},
"outputs": [
@@ -561,7 +615,7 @@
},
{
"cell_type": "code",
"execution_count": 137,
"execution_count": 56,
"id": "fc244f0c",
"metadata": {},
"outputs": [
@@ -594,7 +648,7 @@
},
{
"cell_type": "code",
"execution_count": 138,
"execution_count": 57,
"id": "813e9b68",
"metadata": {},
"outputs": [],
@@ -607,7 +661,7 @@
},
{
"cell_type": "code",
"execution_count": 139,
"execution_count": 58,
"id": "6aa93bec",
"metadata": {},
"outputs": [],
@@ -630,7 +684,7 @@
},
{
"cell_type": "code",
"execution_count": 140,
"execution_count": 59,
"id": "1fb883f8",
"metadata": {},
"outputs": [
@@ -658,40 +712,84 @@
},
{
"cell_type": "code",
"execution_count": 141,
"execution_count": 60,
"id": "a54ce8fb",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"([0.1447573886443752,\n",
" 0.1513815804645033,\n",
" 0.1622363759648628,\n",
" 0.10473845344821518,\n",
" 0.11854956392780605,\n",
" 0.11689825990588008],\n",
" [0.3885783888959868,\n",
" 0.43308922486937373,\n",
" 0.3730220026311381,\n",
" 0.38759443998496274,\n",
" 0.35733354404743983,\n",
" 0.36750626178240353],\n",
" [0.16609258945009867,\n",
" 0.13111313797495522,\n",
" 0.13898110095002836,\n",
" 0.12515895253959083,\n",
" 0.12109768667928446,\n",
" 0.10798315908348614])"
"'sil_orig = [0.1447573886443752, 0.1513815804645033, 0.1622363759648628, 0.10473845344821518, 0.11854956392780605, 0.11689825990588008], sil_pca = [0.3885783888959868, 0.43308922486937373, 0.3730220026311381, 0.38759443998496274, 0.35733354404743983, 0.36750626178240353], sil_sub = [0.16609258945009867, 0.13111313797495522, 0.13898110095002836, 0.12515895253959083, 0.12109768667928446, 0.10798315908348614]'"
]
},
"execution_count": 141,
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sil_orig, sil_pca, sil_sub"
"f\"{sil_orig = }, {sil_pca = }, {sil_sub = }\""
]
},
{
"cell_type": "code",
"execution_count": 61,
"id": "271dc568",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Communities & Crime - Full Metrics Table\n",
"----------------------------------------------------------------------\n",
"k Orig Sil PCA Sil Rand Sil \n",
"----------------------------------------------------------------------\n",
"2 0.1448 0.3886 0.1661 \n",
"3 0.1514 0.4331 0.1311 \n",
"4 0.1622 0.3730 0.1390 \n",
"5 0.1047 0.3876 0.1252 \n",
"6 0.1185 0.3573 0.1211 \n",
"7 0.1169 0.3675 0.1080 \n"
]
}
],
"source": [
"# Full metrics table for all k values - Communities\n",
"print(\"Communities & Crime - Full Metrics Table\")\n",
"print(\"-\" * 70)\n",
"print(f\"{'k':<4} {'Orig Sil':<12} {'PCA Sil':<12} {'Rand Sil':<12}\")\n",
"print(\"-\" * 70)\n",
"for i, k in enumerate(ks):\n",
" print(f\"{k:<4} {sil_orig[i]:<12.4f} {sil_pca[i]:<12.4f} {sil_sub[i]:<12.4f}\")"
]
},
{
"cell_type": "code",
"execution_count": 62,
"id": "d6dcd4f5",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Best PCA k: 3\n",
"Cluster sizes (PCA): [786 253 955]\n",
"Davies-Bouldin (PCA): 0.7937\n",
"Calinski-Harabasz (PCA): 1704.2171\n"
]
}
],
"source": [
"# Best PCA clustering - additional metrics\n",
"best_pca_k = ks[np.argmax(sil_pca)]\n",
"best_pca_labels = labels_pca[best_pca_k]\n",
"\n",
"print(f\"Best PCA k: {best_pca_k}\")\n",
"print(f\"Cluster sizes (PCA): {np.bincount(best_pca_labels)}\")\n",
"print(f\"Davies-Bouldin (PCA): {davies_bouldin_score(Xp, best_pca_labels):.4f}\")\n",
"print(f\"Calinski-Harabasz (PCA): {calinski_harabasz_score(Xp, best_pca_labels):.4f}\")"
]
}
],
lw2/report.md
+117 -31
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@@ -7,34 +7,7 @@ header-includes:
- \usepackage{float}
- \usepackage{tabularx}
- \usepackage{booktabs}
references:
- author:
- family: Redmond
given: Michael
id: communities_and_crime_183
issued: 2002
note: "DOI: https://doi.org/10.24432/C53W3X"
publisher: UCI Machine Learning Repository
title: "[Communities and Crime]{.nocase}"
- author:
- family: Nash
given: Sellers
suffix: Warwick
- family: Ford
given: Wes
id: abalone_1
issued: 1994
note: "DOI: https://doi.org/10.24432/C55C7W"
publisher: UCI Machine Learning Repository
title: Abalone
- author:
- family: Blackard
given: Jock
id: covertype_31
issued: 1998
note: "DOI: https://doi.org/10.24432/C50K5N"
publisher: UCI Machine Learning Repository
title: Covertype
bibliography: "c.bib"
nocite: "[@*]"
---
@@ -52,6 +25,15 @@ We applied K-means clustering to the Abalone dataset. Preprocessing included:
We varied k from 2 to 10, used the **k-means++** centroid initialization, and ran each clustering with multiple initializations (`n_init = 10`) to reduce random seed sensitivity.
### Centroid Initialization: k-means++
The k-means++ algorithm improves upon random initialization by selecting initial centroids that are spread out across the data:
1. **First centroid**: chosen uniformly at random from all data points.
2. **Subsequent centroids**: each new centroid is selected with probability proportional to $D(x)^2$, where $D(x)$ is the distance from point $x$ to the nearest already-chosen centroid.
This weighted sampling ensures centroids are unlikely to start close together, leading to faster convergence and better final solutions compared to naive random initialization.
### Results
#### Silhouette Analysis
@@ -70,6 +52,31 @@ The elbow plot showing inertia (within-cluster sum of squares) across k:
The silhouette score peaked at _k = 2_, indicating that this value of k yields the best separation given the dataset. The elbow plot shows diminishing returns after this point. Overall, the clustering quality suggests moderate structure in the data.
### Dataset Statistics
| Metric | Value |
| ----------------------- | ------------- |
| Samples | 4,177 |
| Features | 9 |
| Best k | 2 |
| Cluster sizes | 1,940 / 2,237 |
| Davies-Bouldin Index | 0.8836 |
| Calinski-Harabasz Index | 4,512.63 |
### Full Results Table
| k | Silhouette | Inertia (WCSS) |
| --- | ---------- | -------------- |
| 2 | **0.431** | 18,066 |
| 3 | 0.331 | 13,419 |
| 4 | 0.345 | 11,157 |
| 5 | 0.333 | 9,331 |
| 6 | 0.374 | 7,660 |
| 7 | 0.392 | 6,764 |
| 8 | 0.391 | 5,940 |
| 9 | 0.386 | 5,315 |
| 10 | 0.391 | 4,805 |
---
## 1.2 Dataset: Covertype [@covertype_31]
@@ -98,6 +105,38 @@ We tested k from 2 to 8, tracked inertia and silhouette scores, and recorded res
The Covertype silhouette scores remained relatively low across k values, indicating weak separation in the original feature space. The elbow plot does not show a clear bend, further suggesting that classic k-means struggles with high dimensionality and feature noise in this dataset.
### Dataset Statistics
| Metric | Value |
| ----------------------- | ---------------------------------------------------------- |
| Samples | 20,000 (subsampled) |
| Features | 54 |
| Best k | 8 |
| Cluster sizes | 1,407 / 6,092 / 3,757 / 2,258 / 1,139 / 4,226 / 88 / 1,033 |
| Davies-Bouldin Index | 2.3425 |
| Calinski-Harabasz Index | 714.62 |
### Full Results Table
| k | Silhouette | Inertia (WCSS) |
| --- | ---------- | -------------- |
| 2 | -0.031 | 1,033,856 |
| 3 | 0.014 | 983,031 |
| 4 | 0.052 | 933,199 |
| 5 | 0.076 | 914,887 |
| 6 | 0.057 | 889,334 |
| 7 | 0.113 | 868,853 |
| 8 | **0.124** | 848,657 |
### Comparison of Datasets
| Dataset | Best k | Silhouette | Davies-Bouldin | Calinski-Harabasz |
| --------- | ------ | ---------- | -------------- | ----------------- |
| Abalone | 2 | 0.431 | 0.88 (better) | 4,513 (better) |
| Covertype | 8 | 0.124 | 2.34 | 715 |
The low silhouette scores for Covertype suggest the dataset may benefit from dimensionality reduction or alternative clustering methods (e.g., DBSCAN for density-based clusters).
---
# Part 2 — Subspace Clustering
@@ -114,6 +153,15 @@ Before clustering:
- Replaced missing values (`?`) with column means.
- Standard scaled all features.
### Dataset Statistics
| Metric | Value |
| --------------------------- | ----- |
| Samples | 1,994 |
| Features | 122 |
| Components for 90% variance | 30 |
| First 2 PCs explain | 35.0% |
## 2.2 PCA Variance Explained
We applied PCA to analyze variance distribution across components and determine an appropriate dimensionality for visualization/clustering.
@@ -145,16 +193,54 @@ For the PCA space, the best silhouette score occurred at _k = 3_. The clusters a
## 2.5 Random Subspace Clustering
We also selected a random subset of 30 features and repeated k-means clustering, comparing silhouette scores with previous methods. This helped assess the impact of arbitrary feature selection vs. PCA-based dimensionality reduction.
We also selected a random subset of 30 features (out of 100+) and repeated k-means clustering. This simulates a naive dimensionality reduction approach without considering feature importance or variance.
### Random Subspace Results
The random subspace silhouette scores are included in the comparison plot above.
### Discussion
#### Full Comparison Table
| k | Original | PCA 2D | Random 30 |
| --- | -------- | --------- | --------- |
| 2 | 0.145 | 0.389 | 0.166 |
| 3 | 0.151 | **0.433** | 0.131 |
| 4 | 0.162 | 0.373 | 0.139 |
| 5 | 0.105 | 0.388 | 0.125 |
| 6 | 0.119 | 0.357 | 0.121 |
| 7 | 0.117 | 0.368 | 0.108 |
#### Best PCA Clustering (k=3)
| Metric | Value |
| ----------------------- | --------------- |
| Cluster sizes | 786 / 253 / 955 |
| Davies-Bouldin Index | 0.7937 |
| Calinski-Harabasz Index | 1,704.22 |
**Key findings:**
- **PCA outperforms random subspace**: By projecting onto directions of maximum variance, PCA preserves cluster structure better than random feature selection.
- **Random subspace is unstable**: Different random seeds would yield different subsets, leading to inconsistent clustering results.
- **Dimensionality reduction is beneficial**: Both methods outperform the original high-dimensional space, confirming that reducing noise dimensions improves cluster separability.
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# Conclusion
We successfully ran k-means clustering on two UCI datasets (Abalone and Covertype). For Abalone, moderate cluster structure was identified, while Covertype presented weak separability in the original space.
We successfully ran k-means clustering on two UCI datasets:
In the subspace clustering section, PCA significantly improved clustering quality relative to both original and random subspace clustering. This demonstrates that **dimensionality reduction can reveal latent structure** that is otherwise obscured in high-dimensional data.
- **Abalone** (4,177 samples, 9 features): Best at k=2 with silhouette 0.431, Davies-Bouldin 0.88, showing moderate cluster structure.
- **Covertype** (20,000 samples, 54 features): Best at k=8 with silhouette 0.124, indicating weak separability due to high dimensionality.
For subspace clustering on **Communities & Crime** (1,994 samples, 122 features):
- PCA to 2D achieved the highest silhouette (0.433 at k=3), a 3× improvement over the original space (0.151).
- Random 30-feature subspace performed worse (0.131 at k=3) than PCA, demonstrating that informed dimensionality reduction outperforms arbitrary feature selection.
- Only 30 PCA components are needed to capture 90% of variance, while the first 2 components explain 35%.
This confirms that **dimensionality reduction can reveal latent cluster structure** otherwise obscured by the curse of dimensionality.
# References
lw2/report.pdf
LFS
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