Kernel (linear algebra)

In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L : V → W between two vector spaces V and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the zero vector in W, or more symbolically: :\ker(L) = \left{ \mathbf{v} \in V \mid L(\mathbf{v})=\mathbf{0} \right} = L^{-1}(0). Properties The kernel of L is a linear subspace of the domain V. In the linear map L : V \to W, two elements of V have the same in W if and only if their difference lies in the kernel of L, that is, L\left(\mathbf{v}_1\right) = L\left(\mathbf{v}_2\right) \quad \text{ if and only if } \quad L\left(\mathbf{v}_1-\mathbf{v}

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The purpose of the course is to introduce the basic notions of linear algebra and its applications.

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Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

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A Determinantal Identity for the Permanent of a Rank 2 Matrix

Adam Wade Marcus

We prove an identity relating the permanent of a rank 2 matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the possibility that these are actually special cases of some more general identity (or class of identities) connecting permanents and determinants. The proof combines some basic facts from the theory of symmetric functions with an application of a famous theorem of Binet and Cauchy in linear algebra.

TAYLOR & FRANCIS INC2022

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Coastal atmospheric temperature prediction in Greenland using support vector regression

Matthew Josef Parkan

In the recent years, global climate change has induced evergrowing loss of sea ice in the Arctic. As the sea ice disappears, albedo diminishes and the sea surface is more likely to be warmed by incoming solar radiation. With the right wind conditions, this extra heat may also be advected towards the shore and thus influence coastal atmospheric temperatures. Thus, knowing how coastal atmospheric temperature is related to offshore conditions is paramount to help predict inshore effects. To study this relation, an exploratory approach using machine learning algorithms is proposed. Based on a combination of daily in situ (i.e. wind velocity, sea level pressure) and remotely sensed (i.e. sea surface temperature, sea ice concentration) data, a series of predicting features are constructed for the years 1981-2010. Two implementations of support vector regression (SVR), one with a linear kernel and the other with a combination of gaussian and histogram intersection kernels are then applied. Results of the SVR indicate that prediction root mean squared errors of less than 5°C are routinely achievable. Prediction errors are also found to be the smallest in summer months and/or at lower latitudes. Finally, the relative importance (ranking) of features appears to be highly variable, depending both on the location and the period of the year.

2012

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Spectral Hypergraph Sparsifiers of Nearly Linear Size

Mikhail Kapralov, Jakab Tardos

Graph sparsification has been studied extensively over the past two decades, culminating in spectral sparsifiers of optimal size (up to constant factors). Spectral hypergraph sparsification is a natural analogue of this problem, for which optimal bounds on the sparsifier size are not known, mainly because the hypergraph Laplacian is non-linear, and thus lacks the linear-algebraic structure and tools that have been so effective for graphs. Our main contribution is the first algorithm for constructing epsilon-spectral sparsifiers for hyper-graphs with O*(n) hyperedges, where O* suppresses (epsilon(-1) log n)(O(1)) factors. This bound is independent of the rank r (maximum cardinality of a hyperedge), and is essentially best possible due to a recent bit complexity lower bound of Omega(nr) for hypergraph sparsification. This result is obtained by introducing two new tools. First, we give a new proof of spectral concentration bounds for sparsifiers of graphs; it avoids linear-algebraic methods, replacing e.g. the usual application of the matrix Bernstein inequality and therefore applies to the (nonlinear) hypergraph setting. To achieve the result, we design a new sequence of hypergraphdependent epsilon-nets on the unit sphere in R-n. Second, we extend the weight-assignment technique of Chen, Khanna and Nagda [FOCS'20] to the spectral sparsification setting. Surprisingly, the number of spanning trees after the weight assignment can serve as a potential function guiding the reweighting process in the spectral setting.

IEEE COMPUTER SOC2022

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