[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124716-en":3,"doc-seo-124716-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},124716,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","The Lattice Overparametrization Paradigm for the Machine Learning of Lattice Operators","Machine learning of lattice operators faces three bottlenecks: statistical design of a constrained low-bias, low-complexity operator class relative to sample size; computation of an efficient empirical-error minimization algorithm; and theoretical understanding of the learned operator’s properties. This work introduces a learning paradigm that overparametrizes operators via elements in a lattice and then learns by applying lattice-based function minimization. The paper presents stochastic lattice descent as a general algorithm once the lattice overparametrization is fixed, discusses prior proof-of-concept results, and shows that operator properties become deducible—improving control, transparency, and interpretability compared with neural-network methods.","arXiv :2310 .06639v2 [ cs .LG] 26 Jan 2024  \nThe Lattice Overparametrization Paradigm for the Machine Learning of Lattice Operators⋆  \nDiego Marcondes 1 ,2[0000−0002−6087−4821] and Junior Barrera 1[0000−0003−0439−0475]  \n1 Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, São Paulo, Brazil.  \n2 Department of Electrical and Computer Engineering, Texas A&M University, College Station, USA  \nAbstract. The machine learning of lattice operators has three possible bottlenecks. From a statistical standpoint, it is necessary to design a constrained class of operators based on prior information with low bias, and low complexity relative to the sample size. From a computational perspective, there should be an efficient algorithm to minimize an empirical error over the class. From an understanding point of view, the properties of the learned operator need to be derived, so its behavior can be theoretically understood. The statistical bottleneck can be overcome due to the rich literature about the representation of lattice operators, but there is no general learning algorithm for them. In this paper, we discuss a learning paradigm in which, by overparametrizing a class via elements in a lattice, an algorithm for minimizing functions in a lattice is applied to learn. We present the stochastic lattice descent algorithm as a general algorithm to learn on constrained classes of operators as long as a lattice overparametrization of it is fixed, and we discuss previous works which are proves of concept. Moreover, if there are algorithms to compute the basis of an operator from its overparametrization, then its properties can be deduced and the understanding bottleneck is also overcome. This learning paradigm has three properties that modern methods based on neural networks lack: control, transparency and interpretability. Nowadays, thereis an increasing demand for methods with these characteristics, and we believe that mathematical morphology is in a unique position to supply them. The lattice overparametrization paradigm could be a missing piece for it to achieve its full potential within modern machine learning.  \nKeywords: lattice overparametrization · discrete morphological neural networks · image processing · mathematical morphology · U-curve algorithms · stochastic lattice descent  \n⋆ Corresponding author: D. Marcondes ([dmarcondes@ime.usp.br](dmarcondes@ime.usp.br)). D. Marcondes was funded by grants \\#22/06211-2 and \\#23/00256-7, São Paulo Research Foundation  \n(FAPESP), and J. Barrera was funded by grants \\#14/50937-1 and \\#2020/06950-4, São Paulo Research Foundation (FAPESP) .  \n2 D. Marcondes and J. Barrera  \n1 Algebraic representations of operators  \nLet (L, ≤) be a complete lattice. A lattice operator ψ : L → L is a mapping from L into itself, and we denote by Ψ = LL the set of all lattice operators in L. The collection Ψ inherits the complete lattice structure of L by considering the pointwise partial order. Let Ω ⊂ Ψ be a complete sublattice of Ψ .  \nAn algebraic representation of (Ω,≤) is any complete lattice (Θ,≤) such that there exists a lattice isomorphism R : Ω → Θ . The element θ ∈ Θ is the parameter that represents the operator ψθ = R −1(θ) and (R,Θ) is a parametrization of Ω . The algebraic representations are not unique and, although they are all equivalent, some have advantages over others.  \nA general algebraic representation of a lattice operator ψ is through its kernel, as proposed in3 [5] . Let ΘK = P (L)L be the collection of all maps F from L to P (L) equipped with the pointwise partial order  \nF1 ≤ F2 ⇐⇒ F1 (Y ) ⊂ F2 (Y ) ∀Y ∈ L for F1 , F2 ∈ ΘK , and consider the lattice isomorphism RK : Ω → ΘK given by  \nRK (ψ)(Y ) = K(ψ)(Y ) = {X ∈ L : Y ≤ ψ (X)} (Y ∈ L) . See [5, Proposition 6.1] for a proof that RK is a lattice isomorphism.  \nThe operators in specific lattices, such as finite lattices, and subclasses of operators in general lattices, such as upper semi-continuo","cbCaiigTRqizkozN","https://ap.wps.com/l/cbCaiigTRqizkozN","pdf",530869,1,13,"English","en",105,"# Algebraic representations of operators\n## Complete lattice and lattice operators\n## Kernel-based algebraic representation\n## Interval basis and sup-generating decomposition\n## Maximal-interval representation mapping\n## Specific operator classes and alternative representations","[{\"question\":\"What are the three main bottlenecks in learning lattice operators?\",\"answer\":\"The paper identifies a statistical bottleneck (designing a constrained class with low bias and complexity), a computational bottleneck (efficient empirical error minimization), and an understanding bottleneck (deriving and explaining properties of the learned operator theoretically).\"},{\"question\":\"How does the proposed paradigm address learning lattice operators?\",\"answer\":\"It overparametrizes a constrained operator class using elements in a lattice, then applies an algorithm for minimizing functions in that lattice to learn the operator.\"},{\"question\":\"What is stochastic lattice descent used for in this framework?\",\"answer\":\"Stochastic lattice descent is presented as a general learning algorithm for constrained classes of lattice operators, provided the lattice overparametrization is fixed.\"}]","The Lattice Overparametrization Paradigm for the Machine Learning of Lattice Operators | 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are the three main bottlenecks in learning lattice operators?","Question",{"text":75,"@type":76},"The paper identifies a statistical bottleneck (designing a constrained class with low bias and complexity), a computational bottleneck (efficient empirical error minimization), and an understanding bottleneck (deriving and explaining properties of the learned operator theoretically).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed paradigm address learning lattice operators?",{"text":80,"@type":76},"It overparametrizes a constrained operator class using elements in a lattice, then applies an algorithm for minimizing functions in that lattice to learn the operator.",{"name":82,"@type":73,"acceptedAnswer":83},"What is stochastic lattice descent used for in this framework?",{"text":84,"@type":76},"Stochastic lattice descent is presented as a general learning algorithm for constrained classes of lattice operators, provided the lattice overparametrization is 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