Technology
Generalizing Rao’s Contributions to Information Geometry in Future Quantum Computing Contexts: A (Very) Preliminary Analysis for Technology Experts.
Dr. Jonathan Kenigson, FRSA
C. R. Rao laid the foundation for modern Information Geometry in 1945 while a research student at Andhra and Calcutta – a task that involved the employment of Manifold Theory in conditions of divergence and entropy. The initial ideas employed in the paradigm involve: (1). Positive-Definite Information Matrices as Norms on Function Spaces; (2). Function Spaces that are restricted to the context(s) of spaces of manifolds of Distribution Functions. In particular, (2) assumes the form of spaces of Normal Distributions, whose global geometry is isomorphic to the “Upper-Half-Plane” Model of Hyperbolic Geometry with the classical Poincaré metric. In a “qualitative” sense, the distinction between two distributions may be interpreted as a proportion of the difference of their relative Information Entropies (IE). This interpretation lends itself well to modern computation and AI because such entropies, while long-known, have just begun to be explored in Quantum contexts. Information from network topologies and architectures explored from categorical contexts are devoid of this expression. In the former paradigm, entropy cannot be quantified in an integrable sense because of the lack of a measure-theoretic expression of the relevant categorical quantities. The Riemannian structure of the Information Manifold, however, renders it a Topological Space, and, more particularly, a “Metric Space,” whose distance function is positive-definite and derived from the concavity conditions implied in the “Fisher-Rao Information Matrix.” Any Information Manifold also possesses a “coherent” integral structure whose “Lebesgue-Measurable” sets are determined canonically by the “Ring of Outer Measures” on the manifold. The resulting Lebesgue Integral has properties that permit efficient computation as guaranteed by abstract Measure Theory – namely, arbitrary pointwise approximation by binary step functions and strong restrictions on information convergence demanded by such results as Fatou’s Lemma and the Lebesgue Dominated Convergence Theorem.
Entropies that generalize the Rényi entropy (like the Kullback-Leibler Divergence), may be constructed on the tangent spaces of an Information Manifold as Surface Integrals. The resulting paradigm permits novel applications of Ergodic metrization. Categorical and Measure-Theoretic notions may be introduced induced through integral Prokhorov metrization and then rendered algebraically by the introduction of Markov Transition Kernels between the manifolds. This approach retains the rich geometry of Rao’s paradigm without sacrificing the expediency and applicability of Categorical notions of transition functions. Given a countable sequence of random variables on a probability space (that are not necessarily identically distributed), one can employ Fisher-Rao theory to construct a sequence of Information Manifolds in the metric tensor defined by the Fisher Information. This manifold sequence is associated with a sequence of spaces of probability measures defined on the Borel subsets induced by each metric. Each of these is capable of Prokhorov Metrization. We introduce a modern measure theory to rigorously define transition kernels between the Fisher-Rao Manifolds, associating a generalized Markov process with each transition.
I present herein a paradigm for this attempt that could permit generalization of existing results to the study of Information Manifolds generated in Quantum Computing – in which, among other differences, the underlying manifolds are not comprised of Normal Distributions but of Wavefunctions from some underlying computational process that generates functional data capable of at least positive-semidefinite metrization in the manner of a Riemannian Manifold. Mathematically, the generalized Kolmogorov Central Limit Theorems are sufficiently robust to permit strong statements about Convergence in Law that generalize the classical CLT to situations in which the underlying random variables are not Independent and Identically Distributed (IID). Relevant resources for each step of the process are presented below. Initially, one begins with the notion of probability as a derivation on the Lebesgue Measures. Even though it is quite old, the Kolmogorov framework is still extremely useful. The best reference for this work available in modern translation is likely still Andrei Nikolajevich Kolmogorov’s 1950 masterwork, Foundations of the Theory of Probability (In Russian). The properties of a random variable on a measurable space are lucidly explored by Bert Fristedt and Lawrence Gray’s 1996 text, A modern approach to probability theory published by Birkhäuser in Boston. Convergence results should encapsulate the measure-theoretic properties of the base space in Lebesgue terms so that a general Smooth Orientable Manifold can be created from local spaces. It is the opinion of the current author that the most lucid account of convergence in such generality is furnished in Billingsley, Patrick (1999). Convergence of probability measures. 2nd ed. John Wiley & Sons. pp.1-28. Once an Information Manifold with a given metric tensor exists, one may set about to explore various divergences induced by: (1). Fisher Information and (2). Quasi-Canonical divergences that exist on the tangent spaces, like the Kullback-Leibler Divergence. It is the current author’s opinion that the best resources devoted to this approach are Nagaoka Hiroshi’s 2000 monograph, Methods of information geometry, Translations of mathematical monographs; v. 191, and Shun’ichi Amari’s 1985 introductory applied text Differential-geometrical methods in statistics published by Springer-Verlag in Berlin. Metrization of the spaces of Borel measures on a sequence of Information Manifolds is possible using so-called Prokhorov Processes. Billingsley’s (1999) treatment is particularly salient because convergence on Riemannian Manifolds also induces convergence on sequences of measures, which are themselves defined on the underlying rings of Sigma Algebras for each manifold’s integral structure.
More abstractly, the current author finds that the Fisher Information can be defined on the tangent space S of arbitrary Radon measures using the Radon-Nikodym Theorem. The clearest and most concise existing resource for this treatment may be found in Mitsuhiro Ito and Yuichi Shishido’s 2008 article, “Fisher information metric and Poisson kernels” in the journal Differential Geometry and Its Applications. 26 (4): 347 – 356. The notion of “Poisson Kernel” is intended in the singular sense of a classical Normal Distribution rather than the sense of Kernel-Superpositions that arise in (for instance) Jacobi Theta representations of pure imaginary arguments. From this vantage, one defines an arbitrary f-Divergence on each Statistical Manifold S(X) in the manner of Coeurjolly, J-F. & Drouilhet, R. (2006). “Normalized information-based divergences”.arXiv:math/0604246. A Categorical view of the Markov Kernels and their transition states may be derived intuitively or read from F. W. Lawvere (1962). “The Category of Probabilistic Mappings” by any available publisher. This resource is free in the public domain in the USA. Further submissions will further elucidate the details of the proposed paradigm.
MongoDB, Inc. (MDB), a Software—Infrastructure corporation, was a bullish stock in the previous trading session. MongoDB, Inc. (MDB) shares opened the market for trading at $158.73 and closed the day at $162.13 with positive move of +1.92%. The increase move in MongoDB, Inc. (MDB) beat the move in the NASDAQ, a tech heavy index, for the day which lossed -0.31% in previous trading session.
MongoDB, Inc. (MDB) is focused on the Software—Infrastructure sector. This stock market sector has seen some action from traders in recent days. Bullish traders of MongoDB, Inc. (MDB) will be looking for positive upcoming earnings whereas bearish traders of MongoDB, Inc. (MDB) will be looking for negative earnings. The last time MongoDB, Inc. (MDB) had an earnings surprise was on Qtr Ending 01/20 when they reported -$0.68 compared to an estimate of -$0.64 which was a difference of -$0.04 resulting in a change of -6.25%. Current earnings estimates are to be released for Current Qtr 04/2020 on MongoDB, Inc. (MDB) and out of 5 Wall Street estimates the average is saying -$0.69. The high estimate is saying -$0.66 whereas the low estimate is saying -$0.72 and the prior year MongoDB, Inc. (MDB) announced earnings at -$0.48 a growth rate est. (year over year) of -43.75%.
The market performance of MongoDB, Inc. (MDB) has varied recently. Year to date MongoDB, Inc. (MDB)’s shares are up 23.19%. Over the past 12 weeks MongoDB, Inc. (MDB) is down -0.90% and over the last 4 weeks MongoDB, Inc. (MDB) is up 29.62%. MongoDB, Inc. (MDB) current income statement shows revenue (ttm) of $421.72M. Gross Profit is $296.36M. This puts the current EBITDA of MongoDB, Inc. (MDB) at $-130.51M.
Wall street traders might also notice recent changes to stock analyst estimates for MongoDB, Inc. (MDB). The stock is trading with an RSI (relative strength index) of 59.71 indicating MongoDB, Inc. (MDB) is neither overbought nor oversold. Technically, MDB’s short term support levels are around $3.54, $2.82 and $1.97 on the downside. MDBs short term resistance levels are $6.00, $5.50 and $5.32 on the upside. The market cap of MongoDB, Inc. (MDB) is $9.333B and institutions hold 99.49% of the stock. The outstanding share count stands at 57.65M while the short float (shares short) stands at 17.56%.
Stock market research shows that these earning estimate changes look to be directly correlated with short-term shares prices. The traders can look to capitalize on the basis of ranking. MongoDB, Inc. (MDB) has short term rating of Bullish (0.33), Intermediate rating of Bullish (0.38) and long-term rating of Neutral (0.06) giving MongoDB, Inc. (MDB) an overall rating of Bullish (0.25).
The EPS growth this year on MongoDB, Inc. (MDB) is decreased -64.90% and is expected to increase 38.30% next year. However, quarterly earnings (EPS) growth year over year on MongoDB, Inc. (MDB) has decreased by -712.40% and quarterly revenue (Sales) growth year over year stands at 28.20%. Gross margin is detected at 70.30% that represents the percent of total sales revenue that the company retains after direct cost of goods sold. Net profit margin (ttm) is -41.60% while return on equity (ttm) is -89.00%.
Recent Developments:
2020-03-17 – Mongodb Reports Q4 Loss Per Share Of $1.10.
2020-02-11 – Mongodb Board Increases Size To Ten Directors – SEC Filing.
2020-01-10 – MongoDB Announces Pricing Of Upsized $1.0 Bln Convertible Senior Notes Offering.
2020-01-08 – Mongodb Announces Proposed Private Offering Of $750 Mln Of Convertible Senior Notes.
2019-12-09 – Mongodb Inc Q3 Non-Gaap Loss Per Share $0.26.
About Company:
MongoDB, Inc. provides general purpose database platform. Its products include MongoDB Enterprise Advanced, MongoDB Enterprise for OEM, MongoDB Professional, MongoDB Stitch, MongoDB Atlas, Development Support, Ops Manager, Cloud Manager, Compass, Connector for business intelligence, and Connector for Spark.
Six years ago, spurred on by a market with an appetite for innovation, major technology ventures were showing the world their glittering smartwatch lines. From that first bake, a few competitors were left standing. The survivors were those teams that today found their reason in an elegant but shock-resistant design, a low energy consumption but able to record the variation of vital signs at the end of the day.
With a 35.8% global share, Apple is the leader in this sector. Counterpoint Research’s report highlights a 49% growth in Apple Watch 4 shipments over the previous model. Samsung ranks second with 11.1% and a 7.2% improvement in the number of units dispatched. Thirdly, the Chinese brand Imoo advanced with 9.2%, followed by Fitbit which advanced with 5.5%. The report numbers global shipments of units to stores between the first quarter of 2018 and the first quarter of 2019.
One of the major differences between these brands’ devices and other smartwatches is the variety of quadrants that can be obtained from their app store, the ease of interchanging their straps, the fluidity provided by the operating system to move around their small screen, and the novel tools for analyzing aspects of body well-being.
Globally, 48% more smartwatches are sold than the previous year, according to research by Counterpoint Research, evidencing a rebound in user interest. Among the top positions, the absence of Android WearOS models is notorious, which shows the little interest that both Google and manufacturers have put in developing this operating system for wearables.
This week the fifth series of Apple Watch was released with the premiere of the operating system WatchOS 6. In terms of digital care, it adds software that measures decibels and warns when harmful sound levels are reached. With the Cycle Tracking app, it allows women to monitor their menstrual cycle, being able to anticipate when their period will arrive.
In addition, if the owner suffers a fall, the device detects it and triggers an alert. If it’s nothing serious and there’s no more than a stumble, it can be canceled. It also comes with the ECG software to perform an electrocardiogram which records the rhythm and intensity of the electrical signals that make the heartbeat.
The most recent local release is Fitbit’s new Versa 2, which offers an indicative sleep quality score based on heart rate, analysis of wake-up times, time spent awake, and sleep phases. The Smart Wake function uses automatic learning to wake the sleeper during light sleep or REM phases, seeking a less aggressive awakening.
Fitbit Versa 2. Three-axis accelerometer, optical heart rate monitor, altimeter, ambient light sensor, relative SpO2 sensor, integrated microphone. 4 days of autonomy. $ 19,500.
Fitbit Versa 2. Three-axis accelerometer, optical heart rate monitor, altimeter, ambient light sensor, relative SpO2 sensor, integrated microphone. 4 days of autonomy. $ 19,500.
In complementary form, it is also possible to obtain an intelligent scale ($ 5.000) that allows to take control of weight and to integrate it together with other data and registers on the level of physical activity or diet. This is the Fitbit Aria Air.
In the Samsung Galaxy Watch Active, the rotating crown that characterized the Galaxy Watch series disappears, making its design much cleaner and clearer. For blood pressure monitoring, you can opt for the My BP Lab application – developed in conjunction with the University of California, San Francisco – to monitor blood pressure and monitor your daily health.
More possibilities open up by linking the watch to the Samsung Health mobile application to seek physical and mental balance with guided deep breathing exercises. It is possible to collect more detailed information about the four stages of sleep, to see if one is getting good quality rest.
Within the formality of the Huawei Watch GT are hidden some surprises. The most striking is its dual-core ARM Cortex M4 chip. One party is responsible for the execution of the basic processes and the other unit is more ambitious, consumes more, but comes into action when more processing capacity is required.
Netflix, Inc. (NFLX), an Entertainment corporation, was a market gainer in the prior market session. Netflix, Inc. (NFLX) shares started the day for trading at $434.14 and closed the day at $435.55 with the declining move of -0.22%. This decrease move lagged the Dow’s +1.91% gain for the prior market day.
Netflix, Inc. (NFLX) is focused on the Entertainment sector. This sector has seen some action from day traders in recent weeks. Bullish day traders of Netflix, Inc. (NFLX) will be looking for positive upcoming earnings whereas bearish day traders of Netflix, Inc. (NFLX) will be looking for negative earnings. The last time Netflix, Inc. (NFLX) had an earnings surprise was on Qtr Ending 03/20 when they reported $1.57 compared to an estimate of $1.61 which was a difference of -$0.04 resulting in a change of -2.48%. Current earnings estimates are to be released for Current Qtr 06/2020 on Netflix, Inc. (NFLX) and out of 10 Wall Street estimates the average is saying $1.81. The high estimate is saying $1.84 whereas the low estimate is saying $1.79 and the prior year Netflix, Inc. (NFLX) announced earnings at $0.60 a growth rate est. (year over year) of +201.67%.
The market performance of Netflix, Inc. (NFLX) has varied recently. Year to date Netflix, Inc. (NFLX)’s shares are up 34.61%. Over the past 12 weeks Netflix, Inc. (NFLX) is up 14.20% and over the last 4 weeks Netflix, Inc. (NFLX) is up 17.49%. Netflix, Inc. (NFLX) current income statement shows revenue (ttm) of $21.4B. Gross Profit is $7.72B. This puts the current EBITDA of Netflix, Inc. (NFLX) at $3.21B.
Wall street day traders might also notice recent changes to stock analyst estimates for Netflix, Inc. (NFLX). The stock is trading with an RSI (relative strength index) of 60.81 indicating Netflix, Inc. (NFLX) is neither overbought nor oversold. Technically, NFLX’s short term support levels are around $1367.66, $1334.96 and $1286.57 on the downside. NFLXs short term resistance levels are $1526.69, $1493.59 and $1432.99 on the upside. The market cap of Netflix, Inc. (NFLX) is $191.557B and institutions hold 83.49% of the stock. The outstanding share count stands at 440.79M while the short float (shares short) stands at 3.58%.
Research indicates that these financial estimate changes look to be directly correlated with short-term shares prices. The day traders can look to capitalize on the basis of ranking. Based on technical analysis, NFLX has short term rating of Neutral (0.17), Intermediate rating of Bullish (0.48) and the long-term rating of Very Bullish (0.58) giving it an overall rating of Bullish (0.41).
The EPS growth this year on Netflix, Inc. (NFLX) is increased 46.90% and is expected to increase 32.67% next year. However, quarterly earnings (EPS) growth year over year on Netflix, Inc. (NFLX) has increased by 105.80% and quarterly revenue (Sales) growth year over year stands at 27.60%. Gross margin is detected at 39.50% that represents the percent of total sales revenue that the company retains after direct cost of goods sold. Net profit margin (ttm) is 10.40% while return on equity (ttm) is 30.80%.
Recent Developments:
2020-04-22 – Netflix Announces Proposed $1 Billion Offering Of Senior Notes.
2020-04-22 – Netflix Says CEO Reed Hastings’ FY 2019 Total Compensation Was $38.6 Mln Versus $36.1 Mln In FY 2018.
2020-04-21 – Netflix Inc reports quarterly earnings per share $1.57.
2020-04-13 – Netflix Strikes First-Look Deal With Comic Book And Graphic Novel Publisher BOOM! Studios.
2020-03-13 – Netflix On Friday Shut Down All Scripted TV And Film Physical Production And Prep For Two Weeks In U.S. And Canada – Hollywood Reporter.
About Company:
Netflix is considered a pioneer in the streaming space. The company evolved from a small DVD-rental provider to a dominant streaming service provider, courtesy of its wide-ranging content portfolio and a fortified international footprint. At the end of first-quarter 2020, the company had 182.
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