From 6f8371f9ff7bb8b26bfe2024e107ce1e686bec62 Mon Sep 17 00:00:00 2001 From: Olivier Blanvillain <olivier.blanvillain@epfl.ch> Date: Sun, 7 Mar 2021 14:00:03 +0100 Subject: [PATCH] Add labs/lab3-k-means --- labs/lab3-k-means/README.md | 265 +++++++++++ labs/lab3-k-means/clusters.png | Bin 0 -> 12372 bytes labs/lab3-k-means/clusters.svg | 290 ++++++++++++ labs/lab3-k-means/points.png | Bin 0 -> 5784 bytes labs/lab3-k-means/points.svg | 270 ++++++++++++ labs/lab3-k-means/start.png | Bin 0 -> 711761 bytes labs/lab3-k-means/step0.png | Bin 0 -> 5917 bytes labs/lab3-k-means/step0.svg | 239 ++++++++++ labs/lab3-k-means/step1.png | Bin 0 -> 8166 bytes labs/lab3-k-means/step1.svg | 257 +++++++++++ labs/lab3-k-means/step2.png | Bin 0 -> 10316 bytes labs/lab3-k-means/step2.svg | 408 +++++++++++++++++ labs/lab3-k-means/step3.png | Bin 0 -> 8556 bytes labs/lab3-k-means/step3.svg | 353 +++++++++++++++ labs/lab3-k-means/step4.png | Bin 0 -> 10080 bytes labs/lab3-k-means/step4.svg | 441 +++++++++++++++++++ labs/lab3-k-means/uniform-random-30steps.png | Bin 0 -> 589823 bytes 17 files changed, 2523 insertions(+) create mode 100755 labs/lab3-k-means/README.md create mode 100755 labs/lab3-k-means/clusters.png create mode 100755 labs/lab3-k-means/clusters.svg create mode 100755 labs/lab3-k-means/points.png create mode 100755 labs/lab3-k-means/points.svg create mode 100644 labs/lab3-k-means/start.png create mode 100755 labs/lab3-k-means/step0.png create mode 100755 labs/lab3-k-means/step0.svg create mode 100755 labs/lab3-k-means/step1.png create mode 100755 labs/lab3-k-means/step1.svg create mode 100755 labs/lab3-k-means/step2.png create mode 100755 labs/lab3-k-means/step2.svg create mode 100755 labs/lab3-k-means/step3.png create mode 100755 labs/lab3-k-means/step3.svg create mode 100755 labs/lab3-k-means/step4.png create mode 100755 labs/lab3-k-means/step4.svg create mode 100644 labs/lab3-k-means/uniform-random-30steps.png diff --git a/labs/lab3-k-means/README.md b/labs/lab3-k-means/README.md new file mode 100755 index 0000000..0bf5dba --- /dev/null +++ b/labs/lab3-k-means/README.md @@ -0,0 +1,265 @@ +# K-Means + +Use the following commands to make a fresh clone of your repository: + +``` +git clone -b kmeans git@gitlab.epfl.ch:lamp/student-repositories-s21/cs206-GASPAR.git cs206-kmeans +``` + +## Useful links + + * [A guide to the Scala parallel collections](https://docs.scala-lang.org/overviews/parallel-collections/overview.html) + * [The API documentation of the Scala parallel collections](https://www.javadoc.io/doc/org.scala-lang.modules/scala-parallel-collections_2.13/latest/scala/collection/index.html) + * [The API documentation of the Scala standard library](https://www.scala-lang.org/files/archive/api/2.13.4) + * [The API documentation of the Java standard library](https://docs.oracle.com/en/java/javase/15/docs/api/index.html) + +**If you have issues with the IDE, try [reimporting the +build](https://gitlab.epfl.ch/lamp/cs206/-/blob/master/labs/example-lab.md#ide-features-like-type-on-hover-or-go-to-definition-do-not-work), +if you still have problems, use `compile` in sbt instead.** + +## Introduction + +In this assignment, you will implement the K-means algorithm for cluster +detection, which is used to partition *n* vectors into *k* clusters. +Here, vectors are separated into clusters based on their mutual similarity -- +vectors that are closer to each other in space are more likely to end up in the +same cluster, and the distant vectors are likely to be in different clusters. +K-means has many applications: it is used in data mining, image filtering and +signal processing. + +Here is a simple example -- let's say that we have a set of vectors in 2D space, +as shown in the following figure: + + + +As a human, you can visually distinguish the three clusters of points in the +image: + + + +When the number of clusters, dimensions and vectors grows, it becomes difficult +and even impossible to manually determine the clusters. +K-means is a simple algorithm that takes a set of vectors (called *points*) and +outputs as set of clusters as follows: + +1. Pick `k` points called *means*. This is called *initialization*. +2. Associate each input point with the *mean* that is closest to it. + We obtain `k` *clusters* of points, and we refer to this process as + *classifying* the points. +3. Update each mean to have the average value of the corresponding cluster. +4. If the `k` means have significantly changed, go back to step 2. + If they did not, we say that the algorithm *converged*. +5. The `k` means represent different clusters -- every point is in the cluster + corresponding to the closest mean. + +Above, two steps need additional discussion. +First, how do we pick the initial `k` means? +The initialization step can be done in many different ways -- we will just +randomly pick some of the input vectors. +Second, how do we know that the algorithm converged? +We will check that, for each mean, the square distance between the old value of +the mean and the new value of the mean is less than or equal to some value +`eta`. + +For a better illustration, here are a few steps of the K-means algorithm. +Initially, we pick a random set of means, shown with "X" in the figure: + + + +Then, we classify the points according to the closest mean ("X"). +The means divide the space into regions, where each point is closer to the +corresponding mean than any other mean -- in the figure, the dotted line depicts +the borders of different regions: + + + +All the points in the same region form one cluster. After having classified the +points, we can update the mean values to the average of all the points in the +cluster: + + + +Each of the means was significantly updated. +This is a good indication that the algorithm did not yet converge, +so we repeat the steps again -- we first classify all the points: + + + +And then we update the means again: + + + +One of the means did not change at all in the last step. +Still, other means have changed so we continue this process until the change +in the position of each point drops below the `eta` value. + +At each iteration of K-means, we can associate multiple points to clusters, +and compute the average of the `k` clusters, in parallel. +Note that the association of a point to its cluster is independent of the +other points in the input, and similarly, the computation of the average of a cluster +is independent of the other clusters. +Once all parallel tasks of the current iteration complete, +the algorithm can proceed to the next iteration. + +K-means is an example of a *bulk synchronous parallel* algorithm (BSP). +BSP algorithms are composed from a sequence of supersteps, each of which +contains: + +- *parallel computation*, in which processes independently perform local + computations and produce some values +- *communication*, in which processes exchange data +- *barrier synchronisation*, during which processes wait until every process + finishes + +Data-parallel programming models are typically a good fit for BSP algorithms, +as each bulk synchronous phase can correspond to some number of data-parallel +operations. + + +## Classifying the points + +In the first part of this assignment, you will classify the input points +according to the square distance to the means. +Input points are described with the following `Point` data-type: + + class Point(val x: Double, val y: Double, val z: Double) + +You will start by implementing the `classify` method: + + def classify(points: ParSeq[Point], means: ParSeq[Point]): ParMap[Point, ParSeq[Point]] + +This method take a sequence of points and a sequence of means, and return +a map collection, which maps each mean to the sequence of points in the corresponding +cluster. + +Hint: Use `groupBy` and the `findClosest` method, which is already defined for +you. After that, make sure that all the means are in the resulting map, even if their +sequences are empty. + + +## Updating the means + +In the second part of this assignment, you will update the means corresponding +to different clusters. + +Implement the method `update`, which takes the map of classified points produced +in the previous step, and the sequence of previous means. +The method returns the new sequence of means: + + def update(classified: ParMap[Point, ParSeq[Point]], oldMeans: ParSeq[Point]): ParSeq[Point] + +Take care to preserve order in the resulting sequence -- the mean `i` in +the resulting sequence must correspond to the mean `i` from `oldMeans`. + +Hint: Make sure you use the `findAverage` method that is predefined for you. + + +## Detecting convergence + +Finally, you will implement convergence detection. +The convergence detection method takes a sequence of old +means and the sequence of updated means, and returns a boolean indicating if the +algorithm converged or not. +Given an `eta` parameter, `oldMeans` and `newMeans`, it returns `true` if the +algorithm converged, and `false` otherwise: + + def converged(eta: Double, oldMeans: ParSeq[Point], newMeans: ParSeq[Point]) + +The algorithm converged iff the square distance between the old and the new mean is less +than or equal to `eta`, for all means. + +Note: the means in the two lists are ordered -- the mean at `i` in `oldMeans` +is the previous value of the mean at `i` in `newMeans`. + +Implement `converged`! + + +## Running the algorithm + +We now have everything we need to run the K-means algorithm. +We only need to combine the previously defined methods in the right way. + +The tail-recursive `kMeans` method takes a sequence of +points `points`, previously computed sequence of means `means`, and the `eta` +value: + + @tailrec final def kMeans(points: ParSeq[Point], + means: ParSeq[Point], eta: Double): ParSeq[Point] + +The `kMeans` method should return the sequence of means, +each corresponding to a specific cluster. + +Hint: `kMeans` implements the steps 2-4 from the K-means pseudocode. + +You can use the following command from within `sbt` to benchmark your code, the +parallel time shoul be significantly faster than the sequential time if your +computer has more than one core: + + > runMain kmeans.KMeansRunner + + +## Use cases + +And now for the fun part -- the K-means algorithm has a lot of use-cases! + +In image processing applications, it can be used to reduce the size of the +color palette, thus compressing the image. This is done by turning a +[true color image](http://en.wikipedia.org/wiki/Color_depth), where each pixel +is encoded into 32 bits, into [indexed color](http://en.wikipedia.org/wiki/Indexed_color), +where each pixel can be encoded with just a few bits. This is done by using k-means to +"cluster" the important colors in the image, thus reducing its palette from +24-bit (`2^24` colors) to just 32 indexed colors, chosen from the 24-bit palette. +Here, pixels from the image are the input vectors, +and their coordinates are the different color channels. + +This is the original true color (24-bit) image: + + + +And this is the indexed color (32 colors) version of it: + + + +So, thanks to your k-means implementation, ScalaShop can now compress images! +You can start ScalaShop by invoking from `sbt`: + + > runMain kmeans.fun.ScalaShop + +The k-means algorithm is very sensitive to the initial choice of means. There +are three choice strategies implemented in ScalaShop: + +* `Uniform Choice` is the simplest strategy. It chooses `n` colors uniformly in +the entire color space, regardless of the colors used in the image. If the image +has a dominant color, the means created by this strategy will likely be very far +away from the clusters formed by this dominant color. You can try setting the +`Uniform Choice` strategy with 1, 10 and 30 steps. You will notice the initial +choice is quite bad, but the quality improves as the k-means algorithm is applied +in more steps. +* `Random Sampling` is another simple strategy, but with better results. For the +initial means, it randomly samples `n` colors from the image. This yields good +results if the image has few dominant colors, but it cannot handle subtle nuances +in the image. Again, if you try this strategy with 1, 10 and 30 k-means iteration +steps, you will notice improvements as the k-means algorithm is ran more. +* `Uniform Random` is the most complex strategy to pick means, but it also produces +the best results. It works by uniformly splitting the color space in sub-spaces. +It then counts the number of pixels that have colors belonging to that sub-space. +Based on this number, it chooses a proportional number of means in the sub-space, +by randomly sampling from the pixels in that sub-space. Therefore, if your image +has dominant colors, this strategy will drop a proportional number of means for +each dominant color, thus allowing the k-means algorithm to capture fine nuances. + +In the EPFL image now available in ScalaShop, the mountains are a good way to see +how well each initial choice of means fares. You also have different strategies +for deciding convergence: + +* `Steps` allows to run a fixed number of steps. After this, the k-means algorithm +is stopped. +* `Eta` corresponds to the means stability, as we showed earlier: if the +means did not move much since the last iteration, the result is considered stable. +* `Sound-to-noise` ratio is a more refined convergence strategy, which does not settle +for stability but tries to minimize the difference between the true color image +and the index color one. This strategy goes beyond `Eta`, but high Sound-to-noise +ratios will prevent the k-means algorithm from finishing! + +With this in mind, enjoy ScalaShop, the ultimate image manipulation tool and +the nice, warm, sunny photo of EPFL! diff --git a/labs/lab3-k-means/clusters.png b/labs/lab3-k-means/clusters.png new file mode 100755 index 0000000000000000000000000000000000000000..1f3bdf3729d54356d8040e2d1e561da35c980ece GIT binary patch literal 12372 zcmeIZhc{f^8#b(uE_!c?GCI+FCkSS=Xh8@@Z=*yXM0AOW=%$1qy3yMtBA#fO5xs<v zFd=#;c<1+hf5p4jd)C_Lth3uXXYaH3ec#u0opg|eF&#A*H4zaJoyh|OD<UFdLqd8? zNkPcTKS=mMxR8bEnb=Sgj#$d4X@q;K;0F$2L_`#$|4YO_I0t?ZGC84!_E75}FKFbG zP*0-B$Vh3QK)*1LC&8Z5L80CyyBb_XL;^%62D&y;r8~=!g|^ctUk>(YN%h(9Fp`NO zDNkKcbIp-M_Q>WZvxOXrr+M~GC3EiSM$Ar<jI8SI98P4y5#Y~B9L)j`DpvUS%2s$% zYdT8fCN%dppKa*AP(Snc+O6yP_QS8y7FJaGwHpJ1YX<{@?=;b2Njs?_49=8jay~NY z@#_obeZeNh{YW=zS<X%>JVh+kSms{ED`!>&z|D)UmZ%nnN;HQlf^_~#$g-28KY(SO zNeby>^RIf^yEY4H#sHTaTC<GjGEFkF0%su_vh?<>Q4A_VXKlBnB5!spyuD_6cQHN~ 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