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    <title>Derek DeSantis on Derek DeSantis</title>
    <link>https://derekdesantis.github.io/</link>
    <description>Recent content in Derek DeSantis on Derek DeSantis</description>
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    <language>en-us</language>
    <copyright>&amp;copy; 2018</copyright>
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    <item>
      <title>Factorizations of Binary Matrices - Rank Relations and the Uniqueness of Boolean Decompositions</title>
      <link>https://derekdesantis.github.io/publication/unique_binary/</link>
      <pubDate>Tue, 22 Dec 2020 00:00:00 -0700</pubDate>
      
      <guid>https://derekdesantis.github.io/publication/unique_binary/</guid>
      <description></description>
    </item>
    
    <item>
      <title>Operator Algebras Generated by Left Invertibles</title>
      <link>https://derekdesantis.github.io/talk/oa-cosy/</link>
      <pubDate>Mon, 25 May 2020 00:00:00 -0600</pubDate>
      
      <guid>https://derekdesantis.github.io/talk/oa-cosy/</guid>
      <description>&lt;p&gt;Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras.   The Toeplitz algebra is a classic example. The generator $M_z$ on the Hardy space $H^2(\mathbb{T})$ is isometric, and hence, left invertible. Motivated by questions from linear equations in Hilbert spaces (frame theory), we wish to understand particular types of operator algebras generated by left invertible operators.&lt;/p&gt;

&lt;p&gt;In this talk, we investigate the norm closed operator algebra generated by a single left invertible operator $T$ in $\mathscr{B}(\mathscr{H})$ with a canonical left inverse (the Moore-Penrose inverse). This algebra, has rich operator algebraic structure that mimics the Toeplitz algebra. We are particularly interested in a class of left invertible operators we call \textit{analytic}. The adjoints of analytic left invertible operators are Cowen-Dogulas, an important class of operators that serve as a bridge between operator theory and complex geometry. Consequently, we can represent $T$ as $M_z$ on a reproducing kernel Hilbert space of analytic functions. Using tools of Cowen-Douglas theory, we reduce the challenge of classifying these algebras into more tractable geometric problems.&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>Robust and interpretable unsupervised machine learning techniques for analyzing the climate system</title>
      <link>https://derekdesantis.github.io/talk/climate-cluster/</link>
      <pubDate>Thu, 05 Mar 2020 00:00:00 -0700</pubDate>
      
      <guid>https://derekdesantis.github.io/talk/climate-cluster/</guid>
      <description></description>
    </item>
    
    <item>
      <title>Coarse-Grain Cluster Analysis of Tensors With Application to Climate Biome Identification</title>
      <link>https://derekdesantis.github.io/talk/coarse-grain-clustering-coda/</link>
      <pubDate>Tue, 25 Feb 2020 00:00:00 -0700</pubDate>
      
      <guid>https://derekdesantis.github.io/talk/coarse-grain-clustering-coda/</guid>
      <description></description>
    </item>
    
    <item>
      <title>Machine Learning Safety with Applications to the Climate Sciences</title>
      <link>https://derekdesantis.github.io/talk/coarse-grain-clustering-cnls/</link>
      <pubDate>Thu, 06 Feb 2020 00:00:00 -0700</pubDate>
      
      <guid>https://derekdesantis.github.io/talk/coarse-grain-clustering-cnls/</guid>
      <description></description>
    </item>
    
    <item>
      <title>Coarse-Grain Cluster Analysis of Tensors With Application to Climate Biome Identification</title>
      <link>https://derekdesantis.github.io/publication/cgc/</link>
      <pubDate>Tue, 21 Jan 2020 00:00:00 -0700</pubDate>
      
      <guid>https://derekdesantis.github.io/publication/cgc/</guid>
      <description></description>
    </item>
    
    <item>
      <title>Multiresolution Cluster Analysis—Addressing Trust in Climate Classification</title>
      <link>https://derekdesantis.github.io/talk/coarse-grain-clustering/</link>
      <pubDate>Wed, 15 Jan 2020 00:00:00 -0700</pubDate>
      
      <guid>https://derekdesantis.github.io/talk/coarse-grain-clustering/</guid>
      <description></description>
    </item>
    
    <item>
      <title>Nonnegative Canonical Polyadic Decomposition with Rank Deficient Factors</title>
      <link>https://derekdesantis.github.io/publication/ncpd-rk-deficient/</link>
      <pubDate>Tue, 17 Sep 2019 00:00:00 -0600</pubDate>
      
      <guid>https://derekdesantis.github.io/publication/ncpd-rk-deficient/</guid>
      <description></description>
    </item>
    
    <item>
      <title>Operator Algebras Generated by Left Invertibles</title>
      <link>https://derekdesantis.github.io/talk/oa-left-inv-talk-defense/</link>
      <pubDate>Wed, 13 Mar 2019 00:00:00 -0600</pubDate>
      
      <guid>https://derekdesantis.github.io/talk/oa-left-inv-talk-defense/</guid>
      <description>&lt;p&gt;Operator algebras generated by partial isometries and their adjoints form the basis for some of the most well studied classes of C*-algebras. Representations of such algebras encode the dynamics of orthonormal sets in a Hilbert space. We instigate a research program on concrete operator algebras that model the dynamics of Hilbert space frames.&lt;/p&gt;

&lt;p&gt;The primary object of this thesis is the norm-closed operator algebra generated by a left invertible T together with its Moore-Penrose inverse. Of particular interest is when T satisfies a nondegeneracy condition called analytic. We show that T is analytic if and only if its adjoint belongs to an important class of operators called Cowen-Douglas. When the Fredholm index of T is -1, the algebra contains the compact operators, and any two such algebras are boundedly isomorphic if and only if they are similar. A classification of these algebras by K-theoritc means is then acquired.&lt;/p&gt;
</description>
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    <item>
      <title>Operator Algebras Generated by Left Invertibles</title>
      <link>https://derekdesantis.github.io/publication/thesis/</link>
      <pubDate>Fri, 01 Mar 2019 00:00:00 -0700</pubDate>
      
      <guid>https://derekdesantis.github.io/publication/thesis/</guid>
      <description></description>
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    <item>
      <title>Operator Algebras Generated by a Single Left Invertible</title>
      <link>https://derekdesantis.github.io/publication/oa-gen-left-inf/</link>
      <pubDate>Mon, 01 Oct 2018 00:00:00 -0600</pubDate>
      
      <guid>https://derekdesantis.github.io/publication/oa-gen-left-inf/</guid>
      <description></description>
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    <item>
      <title>Example Page</title>
      <link>https://derekdesantis.github.io/tutorial/example/</link>
      <pubDate>Sun, 09 Sep 2018 00:00:00 -0600</pubDate>
      
      <guid>https://derekdesantis.github.io/tutorial/example/</guid>
      <description>

&lt;p&gt;In this tutorial, I&amp;rsquo;ll share my top 10 tips for getting started with Academic:&lt;/p&gt;

&lt;h2 id=&#34;tip-1&#34;&gt;Tip 1&lt;/h2&gt;

&lt;p&gt;&amp;hellip;&lt;/p&gt;

&lt;h2 id=&#34;tip-2&#34;&gt;Tip 2&lt;/h2&gt;

&lt;p&gt;&amp;hellip;&lt;/p&gt;
</description>
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    <item>
      <title>Climate Biome Clustering</title>
      <link>https://derekdesantis.github.io/project/climate-clustering/</link>
      <pubDate>Wed, 27 Apr 2016 00:00:00 -0600</pubDate>
      
      <guid>https://derekdesantis.github.io/project/climate-clustering/</guid>
      <description>&lt;p&gt;There are many factors that determine how we perceive and ultimately classify the climate into distinct regions, or biomes.  For example, precipitation, temperature, flora and fauna are key ingredients in our delineation of biomes. Classifying biomes is akin to understanding the homogeneity of climate and resources on a local scale.  It is not only important that we can classify climate biomes, but also their evolution with the changing climate.  For emphasis, it is not possible to perform any meaningful environmental preservation without predetermined knowledge of what the environment is at a coarse scale.&lt;/p&gt;

&lt;p&gt;Historically, the standard used to classify climate biomes has been the Köppen-Geiger (KG) model [1].  The KG model is an expert based judgment that describes climate zones based off of temperature and precipitation measurements.   The model utilizes a fixed decision tree, where each branch uses various information about temperature and precipitation. This heuristic allows one to broadly assess climate regions.  While KG is interpretable, it is overly simplistic and somewhat arbitrary.&lt;/p&gt;

&lt;p&gt;A solution to this problem is to move towards data-driven methods of classification.  Various authors have noted this, and have applied unsupervised models to classify the climate (see for example, [2], [3]).  These data driven approaches to climate clustering are an epistemological improvement over the use chosen heuristics of KG.  However, they still suffer from two key problems.  First, the algorithms themselves are user chosen, and therefore somewhat arbitrary themselves. There is no external measurement to measure how much one trusts the clustering result. Second, because the algorithms are dependent on the input data, they are dependent on the scale at which the data is acquired.  A priori, it is not clear how this &amp;ldquo;hidden parameter&amp;rdquo; of scale affects the overall clustering result. The goal of this project is to address both of these problems.&lt;/p&gt;

&lt;p&gt;[1] Markus Kottek, Jürgen Grieser, Christoph Beck, Bruno Rudolf, and Franz Rubel. World map of the köppen-geigerclimate classification updated.Meteorologische Zeitschrift, 15(3):259–263, 2006.&lt;/p&gt;

&lt;p&gt;[2] Pawel Netzel and Tomasz Stepinski. On using a clustering approach for global climate classification.Journal ofClimate, 29(9):3387–3401, 2016.&lt;/p&gt;

&lt;p&gt;[3] Jakob Zscheischler, Miguel D Mahecha, and Stefan Harmeling. Climate classifications: the value of unsupervisedclustering.Procedia Computer Science, 9:897–906, 2012.&lt;/p&gt;
</description>
    </item>
    
    <item>
      <title>Operator Algebras Modeling Dynamics of Frames</title>
      <link>https://derekdesantis.github.io/project/oa-from-frame/</link>
      <pubDate>Wed, 27 Apr 2016 00:00:00 -0600</pubDate>
      
      <guid>https://derekdesantis.github.io/project/oa-from-frame/</guid>
      <description>&lt;p&gt;For a Hilbert space $\mathscr{H}$, $y \in \mathscr{H}$ and  $T \in \mathscr{B}(\mathscr{H})$, the equation
\begin{equation}
T x = y
\end{equation}
has a solution whenever $T$ is left invertible.  For any left inverse $L$ of $T$, $x = L y$ is a solution.  More generally, if $T$ has closed range, $T$ has a &lt;em&gt;pseudo left inverse&lt;/em&gt;. Concretely, if $T \in \mathscr{B}(\mathscr{H})$ has closed range, restricting to $\ker(T)^\perp$ we obtain a new operator that is left invertible. Picking a left inverse $L$ for $T\mid_{\ker(T)^\perp}$, we find that $x = L y$ is a solution.&lt;/p&gt;

&lt;p&gt;In most applications, there is a canonical choice for the left inverse $L$.  Generally, this is the &lt;em&gt;Moore-Penrose inverse&lt;/em&gt;, denoted $T^\dagger$.  It is the unique left inverse such that $\ker(T^\dagger) = \mbox{ran}(T)^\perp$.  To construct $T^\dagger$, one considers $T$ mapping $\ker(T)^\perp$ to $\mbox{ran}(T)$.  From this perspective, $T$ is bijective, and thus boundedly invertible by the open mapping theorem. The unique inverse, once extended by zero, is $T^\dagger$. For instance, if $T$ is an isometry then $T^\dagger = T^*$, the adjoint of $T$.&lt;/p&gt;

&lt;p&gt;One subject where closed range operators are frequently used is &lt;em&gt;frame theory&lt;/em&gt;. A sequence {$f_n$} in a Hilbert space $\mathscr{H}$ is said to be a &lt;em&gt;frame&lt;/em&gt; if there exists constants $0&amp;lt; A \leq B$ such that
\begin{equation}
A ||x||^2 \leq \sum_n | \langle x, f_n \rangle |^2 \leq B ||x||^2
\end{equation}
for all $x \in \mathscr{H}$.&lt;/p&gt;

&lt;p&gt;Associated to each Hilbert space frame {$f_n$} is a (canonical) dual frame {$g_n$}. Using this dual frame, one can reconstruct elements $f$ of the Hilbert space $\mathscr{H}$ in an analogous way to orthonormal basis:&lt;/p&gt;

&lt;p&gt;\begin{equation}
f = \sum_{n \geq 1} \langle f, g_n \rangle f_n
\end{equation}&lt;/p&gt;

&lt;p&gt;It is easy to see that orthonormal bases are frames, but not all frames need be orthogonal, norm one, or even contain a unique set of elements.  A frame does not enforce the rigidity of inner products that an orthonormal basis does - allowing for variation between individual frame elements (rather than just 0 or 1). The flexibility of the definition has engendered applications across signal processing and harmonic analysis. For instance, frames may be constructed for particular features of a problem that are precluded from orthogonal bases due to their requirement for linear independence.  Further, this extra redundancy helps to protect signals from degradation, ensuring that the effects of erasures are minimized.  The looseness of the structure allows one to construct the analog of frames for structures that don&amp;rsquo;t necessarily come equipped with suitable generalization of an orthonormal basis.  Indeed, certain classes of Hilbert C*-Modules and Banach spaces posses frames.&lt;/p&gt;

&lt;p&gt;Many questions concerning Hilbert space frames are inexorably tied to questions of closed ranged and left invertible operators.  All frames for closed subspaces of a Hilbert $\mathscr{H}$ spaces arise as the application of a closed range operator $T$ on a frame for $\mathscr{H}$. Properties of frames, such as rigidity and perturbation results also depend on the analysis of closed range operators.  With this groundwork of closed ranged operators and frame theory laid, we now describe our interest in generating concrete operator algebras.&lt;/p&gt;

&lt;p&gt;A well-studied class of operator algebras are C&lt;em&gt;-algebras generated by partial isometries.    For example graph algebras, C&lt;/em&gt;-algebras generated partial isometries subject to constraints of a directed graph, reside here.&lt;/p&gt;

&lt;p&gt;Representations of graph C*-algebras arising from a directed graph may be described as follows. Given a directed graph, choose a Hilbert space for each vertex of the graph (with suitable dimension depending on loops of the graph).  After direct summing each of these spaces together, we obtain a Hilbert space $\mathscr{H}$ where each vertex corresponds to an orthogonal summand of $\mathscr{H}$.  By choosing orthonormal sequences for each of these closed, orthogonal subspaces of $\mathscr{H}$, we then pick partial isometries that map one summand to another, subject to the Cuntz-Krieger relations.&lt;/p&gt;

&lt;p&gt;The partial isometric generators of concrete graph algebras preserve orthonormal basis (that is, up to the kernels of the partial isometries).  The representation of the graph C&lt;em&gt;-algebra can be viewed as encoding the dynamics of walks on the graph.  A natural extension of these concrete graph C&lt;/em&gt;-algebras is one that encodes the dynamics of a frame.&lt;/p&gt;

&lt;p&gt;One arrives at a such an extension by replacing &lt;em&gt;partial isometries and their adjoints&lt;/em&gt; with &lt;em&gt;closed ranged operators and their Moore-Penrose inverses&lt;/em&gt;.  As discussed above, the closed range operators  preserve frame theoretic quantities.  In the isometric case, the Moore-Penrose inverse is the adjoint. Hence, such operator algebras reduce to the C&lt;em&gt;-algebraic case when the generators are isometric. Therefore, this class of operator algebras is the natural generalization of concrete graph C&lt;/em&gt;-algebras that integrates frame theory.  This leads us to the following general program:&lt;/p&gt;

&lt;p&gt;&lt;em&gt;Given a set of operators with closed range and their Moore-Penrose inverses construct the norm-closed algebra subject to the constraints of a directed graph. What is the structure of these algebras?&lt;/em&gt;&lt;/p&gt;
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    <item>
      <title>Tensor Factorizations of Southern Ocean</title>
      <link>https://derekdesantis.github.io/project/tensor-factor-so/</link>
      <pubDate>Wed, 27 Apr 2016 00:00:00 -0600</pubDate>
      
      <guid>https://derekdesantis.github.io/project/tensor-factor-so/</guid>
      <description>&lt;p&gt;Understanding the climate system is difficult given its dependence on a wide range of length and timescales.  Analyzing Earth system  model (ESM) data, such as that from the Energy Exascale Earth System Model (E3SM), is further complicated  due to the complex interactions between different components of the Earth’s system model. Indeed, most of climate science focuses on single component analysis, e.g. ocean or sea ice. At present, ESMs are constructed to understand average statistics, such as total conservation of heat and moisture.  However, sophisticated dynamics such as heat and moisture transport in coupled system simulations remains elusive. Discovering sophisticated latent climatic signals in fully coupled E3SM data is a challenging task for a number of reasons. Principal among them, we currently do not fully understand the latent signatures within each component, or how these signals will be amplified when information is shared between components.&lt;/p&gt;

&lt;p&gt;In this project, we developed an interpretable unsupervised machine learning technique for diagnostics of the E3SM MPAS-O model. Here, we have temperature and salinity data round The Southern Ocean.  Depending on the length of the simulation, one can incur model discrepancies. Ideally we would like to automatically identify physically relevant instabilities within the model.  Pinpointing how key water masses evolve with time can help provide these insights.
We preform an unsupervised clustering technique to separate the data into classes (Gaussian mixture model).  We then analyze where these classes reside by depth. By looking at key overlapping depth regions, one can interpret the vertical mixing and identify which clusters belongs to the important water masses. Comparing the water masses across two time periods provides insight into how the model has drifted over time.  These insights are leveraged alongside traditional nonnegative tensor factorization techniques to discover latent climate signatures.&lt;/p&gt;
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