Las demás facultades que le confieren otras leyes.”

In higher dimensions 𝑑 ≥ 4, two main methods are considered to identify the convex hull: 1- gift wrapping method [5, 66] 2- beneath-beyond method [63].

The idea behind the gift wrapping method is constructing the convex hull by starting a facet and finding the adjacent facets. This procedure is iteratively conducted for each new identified facet. Since each facet is linked to its corresponding adjacent facets, the way in which the convex hull formed is like wrapping around a convex polytope in 𝑑-dimensional space. Jarvis’s march algorithm introduced in Section 3.3.2 is a special case of the gift wrapping method in a 2-dimensional space. Like Jarvis’s march, the gift wrapping method in high dimensions is an output-sensitive method whose time complexity also depends on the size of output 𝑓 which can be the number of facets of the generated convex hull. Therefore, the time complexity of the gift wrapping method in high dimensions is 𝑂(𝑛𝑓). Based on the upper bound theory, the number of generated facets is 𝑂(𝑣⌊𝑑2⌋) where 𝑣 is the number of

convex hull vertices. In the worst case where 𝑛 = 𝑣, the time complexity of the gift wrapping method is 𝑂(𝑛⌊𝑑2⌋+1). An improved version of the original gift wrapping method was proposed

by Seidel [66] where the algorithm takes 𝑂(𝑛2_{+ 𝑓 log 𝑛) time. Based on the upper bound}

theory, in the worst case, the time complexity of the algorithms is 𝑂(𝑛⌊𝑑2⌋log 𝑛).

The Beneath-beyond method is considered as an incremental approach constructing the convex hull by adding one point into the current convex hull at a time. The update process of the current convex hull includes adding new facets into the current convex hull and removing the visible facets with respect to the new point. The Quickhull algorithm, as a deterministic incremental algorithm stated in Section 3.3.3, and also the randomized incremental algorithms, described in Section 3.3.7, where the convex hull is presented in terms of vertices (i.e., V-representation) are special beneath-beyond methods in a 2-dimensionl Euclidean space. The time complexity of the beneath-beyond method in high dimensions is 𝑂(𝑛 log 𝑛 + 𝑛⌊(𝑑+1)2 ⌋) with 𝑂(𝑛⌊

𝑑

2⌋) space [63]. The method was improved by [67] through derandomizing

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improved version is 𝑂(𝑛 log 𝑛 + 𝑛⌊𝑑2⌋). Since both time and memory requirement increase

exponentially with respect to dimension 𝑑 ≥ 4 with a fixed number of samples, applying the traditional convex hull algorithms in very high dimensions with a huge number of samples is not feasible.

As the state of the art, one of the recent work done to overcome these challenges in high dimensions is the proposal of an approximation convex hull algorithm by Wangs and et al. [68]. The proposed algorithm represents the convex hull in terms of vertices with the aim of including extreme points in the training set for online adaptation process of SVM models. The algorithm is based on samples partitioning where for each partition, the corresponding sub- convex hull is obtained and then the union of vertices of all sub-convex hulls is considered as the set of vertices of an approximation convex hull. The algorithm results an approximation convex hull throughout three steps. In the first step, 𝑑 + 1 samples are selected as the vertices of the initial convex hull so that these samples can constitute a 𝑑-simplex as large as possible where 𝑑 is the dimension. Since the 𝑑-simplex has 𝑑 + 1 facets, it divides the space into 𝑑 + 1 partitions. For example, Fig. 3.10 illustrates a 2-simplex which is translated into a triangle. As it can be seen in Fig. 3.10, the vertices of the 2-simplex are {𝑥1, 𝑥2, 𝑥3}. Assume 𝑜

is the center of the 2-simplex. As can be seen in Fig. 3.10, the rays 𝑜𝑥₁, 𝑜𝑥₂ and 𝑜𝑥₃ divide the samples outside the 2-simplex into three partitions named 𝑃1, 𝑃2 and 𝑃3.

Fig. 3.10. Constructing a large 2-simplex

In the second step, each partition whose number of samples is greater than a user-defined threshold 𝐿 is divided into 𝑑 new partitions based on the furthest sample to the corresponding facet of the partition. Afterwards, the furthest sample is appended into the list of convex hull

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vertices. This procedure continues until there exists no partition whose number of samples exceeds 𝐿. For example, in Fig. 3.11, partition 𝑃3 is divided into two new smaller partitions

named 𝑃31 and 𝑃32 based on the furthest sample named 𝑥4 to the facet 𝑥2𝑥3. As it can be seen

in Fig. 3.11, the facet 𝑥₂𝑥₃ is removed and two new facets 𝑥₂𝑥₄ and 𝑥₃𝑥₄ are generated.

Fig. 3.11. Partitioning step

At the end of this step, each facet is considered as a sub-convex hull including 𝑑 vertices. In the third step, each sub-convex hull is tried to be expanded by identifying the furthest sample to the whole sub-convex hull and appending it into the list of the sub-convex hull vertices. This procedure continues until an approximation convex hull with at most 𝑀 vertices is obtained, where 𝑀 is a user-defined threshold. The time complexity of the algorithm is at most 𝑂(𝑛𝑑4_{) where 𝑛 is the number of samples. Although the algorithm can cope with the}

time complexity in high dimensions, as it will be shown in Section 4.4.1, it presents some points as vertices of the approximation convex hull that do not belong to the vertices of the corresponding real convex hull.