scholarly journals Molecular spaces and the dimension paradox

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ramon Carbó-Dorca
Keyword(s):  
Linear Independence ◽  
Vector Representations

Abstract In this study, the dimension or dimensionality paradox is defined and discussed in a dedicated context. This paradox appears when discrete vector representations of the elements of a molecular set are constructed employing several descriptor parameters, adopting specific values for each molecule. The dimension paradox consists in that when constructing discrete N-dimensional molecular vectors, the primal structure of the attached molecular set, chosen as a collection of different objects, cannot be well-defined if the number of descriptors N and the number of molecules M do not bear a convenient relation like: N ≥ M $N\ge M$ . This has implications for the linear independence of the vectors connected with each molecule.

Mol2vec: Unsupervised Machine Learning Approach with Chemical Intuition

2017 ◽  
Author(s):  
Sabrina Jaeger ◽  
Simone Fulle ◽  
Samo Turk
Keyword(s):  
Machine Learning ◽  
Language Processing ◽  
Supervised Machine Learning ◽  
Learning Approach ◽  
Learning Approaches ◽  
Unsupervised Machine Learning ◽  
Feature Representations ◽  
Machine Learning Approach ◽  
The Individual ◽  
Vector Representations

Inspired by natural language processing techniques we here introduce Mol2vec which is an unsupervised machine learning approach to learn vector representations of molecular substructures. Similarly, to the Word2vec models where vectors of closely related words are in close proximity in the vector space, Mol2vec learns vector representations of molecular substructures that are pointing in similar directions for chemically related substructures. Compounds can finally be encoded as vectors by summing up vectors of the individual substructures and, for instance, feed into supervised machine learning approaches to predict compound properties. The underlying substructure vector embeddings are obtained by training an unsupervised machine learning approach on a so-called corpus of compounds that consists of all available chemical matter. The resulting Mol2vec model is pre-trained once, yields dense vector representations and overcomes drawbacks of common compound feature representations such as sparseness and bit collisions. The prediction capabilities are demonstrated on several compound property and bioactivity data sets and compared with results obtained for Morgan fingerprints as reference compound representation. Mol2vec can be easily combined with ProtVec, which employs the same Word2vec concept on protein sequences, resulting in a proteochemometric approach that is alignment independent and can be thus also easily used for proteins with low sequence similarities.

Examples of theories of presheaf type

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Finite Groups ◽  
Linear Independence ◽  
Geometric Theory ◽  
Vector Spaces ◽  
Linear Orders ◽  
Locally Finite ◽  
Strong Unit ◽  
Locally Finite Groups ◽  
Simplicial Sets ◽  
Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

Scholar2vec: Vector Representation of Scholars for Lifetime Collaborator Prediction

2021 ◽  
Vol 15 (3) ◽  
pp. 1-19
Author(s):  
Wei Wang ◽  
Feng Xia ◽  
Jian Wu ◽  
Zhiguo Gong ◽  
Hanghang Tong ◽  
...  
Keyword(s):  
Scientific Collaboration ◽  
Early Stage ◽  
Collaboration Network ◽  
Vector Representation ◽  
Network Embedding ◽  
Machine Learning Methods ◽  
Academic Networks ◽  
Special Relationships ◽  
Real World Datasets ◽  
Vector Representations

While scientific collaboration is critical for a scholar, some collaborators can be more significant than others, e.g., lifetime collaborators. It has been shown that lifetime collaborators are more influential on a scholar’s academic performance. However, little research has been done on investigating predicting such special relationships in academic networks. To this end, we propose Scholar2vec, a novel neural network embedding for representing scholar profiles. First, our approach creates scholars’ research interest vector from textual information, such as demographics, research, and influence. After bridging research interests with a collaboration network, vector representations of scholars can be gained with graph learning. Meanwhile, since scholars are occupied with various attributes, we propose to incorporate four types of scholar attributes for learning scholar vectors. Finally, the early-stage similarity sequence based on Scholar2vec is used to predict lifetime collaborators with machine learning methods. Extensive experiments on two real-world datasets show that Scholar2vec outperforms state-of-the-art methods in lifetime collaborator prediction. Our work presents a new way to measure the similarity between two scholars by vector representation, which tackles the knowledge between network embedding and academic relationship mining.

scholarly journals Convergence properties of the Broyden-like method for mixed linear–nonlinear systems of equations

2021 ◽  
Author(s):  
Florian Mannel
Keyword(s):  
Linear Independence ◽  
Affine Subspace ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Convergence Properties ◽  
Initial Matrix ◽  
First Time ◽  
Special Case ◽  
Dimensional Mapping ◽  
Finite Number Of Iterations

AbstractWe consider the Broyden-like method for a nonlinear mapping $F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ F : ℝ n → ℝ n that has some affine component functions, using an initial matrix B0 that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping $G:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ G : ℝ d → ℝ d , where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B0, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by $\frac {\sqrt {5}+1}{2}$ 5 + 1 2 if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

Student attitudes towards linear algebra: an attempt to roll back the fog

2020 ◽  
Author(s):  
Barry J Griffiths ◽  
Samantha Shionis
Keyword(s):  
Student Perceptions ◽  
Linear Algebra ◽  
Student Attitudes ◽  
Linear Independence ◽  
Negative Emotions ◽  
Practical Applications ◽  
Key Concepts ◽  
The Subject ◽  
Roll Back

Abstract In this study, we look at student perceptions of a first course in linear algebra, focusing on two specific aspects. The first is the statement by Carlson that a fog rolls in once abstract notions such as subspaces, span and linear independence are introduced, while the second investigates statements made by several authors regarding the negative emotions that students can experience during the course. An attempt is made to mitigate this through mediation to include a significant number of applications, while continually dwelling on the key concepts of the subject throughout the semester. The results show that students agree with Carlson’s statement, with the concept of a subspace causing particular difficulty. However, the research does not reveal the negative emotions alluded to by other researchers. The students note the importance of grasping the key concepts and are strongly in favour of using practical applications to demonstrate the utility of the theory.

Polarity classification using structure-based vector representations of text

2015 ◽  
Vol 74 ◽  
pp. 46-56 ◽  
Author(s):  
Alexander Hogenboom ◽  
Flavius Frasincar ◽  
Franciska de Jong ◽  
Uzay Kaymak
Keyword(s):  
Polarity Classification ◽  
Vector Representations

scholarly journals A converse to linear independence criteria, valid almost everywhere

2015 ◽  
Vol 38 (3) ◽  
pp. 513-528 ◽  
Author(s):  
S. Fischler ◽  
M. Hussain ◽  
S. Kristensen ◽  
J. Levesley
Keyword(s):  
Linear Independence ◽  
Almost Everywhere
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