Fundamental matrix (computer vision)

In mathematics, a Bregman divergence or Bregman distance is similar to a metric, but does not satisfy the triangle inequality nor symmetry. There are two ways in which Bregman divergences are important. Firstly, they generalize squared Euclidean distance to a class of distances that all share similar properties. Secondly, they bear a strong connection to exponential families of distributions; as has been shown by (Banerjee et al. 2005), there is a bijection between regular exponential families and regular Bregman divergences.

Bregman divergences are named after L. M. Bregman, who introduced the concept in 1967. More recently researchers in geometric algorithms have shown that many important algorithms can be generalized from Euclidean metrics to distances defined by Bregman divergence (Banerjee et al. 2005; Nielsen and Nock 2006; Boissonnat et al. 2010).

Definition

Let $F:\mathrm{\Omega }\to \mathbb{ℝ}$ be a continuously-differentiable real-valued and strictly convex function defined on a closed convex set $\mathrm{\Omega }$.

The Bregman distance associated with F for points $p,q\in \mathrm{\Omega }$ is the difference between the value of F at point p and the value of the first-order Taylor expansion of F around point q evaluated at point p:

$D_F^q(p,q)=F(p)-F(q)-⟨\mathrm{\nabla }F(q),p-q⟩.$

Properties

• Non-negativity: $D_F^q(p,q)\ge 0$ for all p, q. This is a consequence of the convexity of F.
• Convexity:$D_F^q(p,q)$ is convex in its first argument, but not necessarily in the second argument (see )
• Linearity: If we think of the Bregman distance as an operator on the function F, then it is linear with respect to non-negative coefficients. In other words, for $F_1,F_2$ strictly convex and differentiable, and $\lambda \ge 0$,

$D_{F_1+\lambda F_2}^q(p,q)=D_{F_1}^q(p,q)+\lambda D_{F_2}^q(p,q)$

• Duality: The function F has a convex conjugate $F^{*}$. The Bregman distance defined with respect to $F^{*}$ has an interesting relationship to $D_F^q(p,q)$

$D_{F^{*}}^{q^{*}}(p^{*},q^{*})=D_F^p(q,p)$

Here, $p^{*}=\mathrm{\nabla }F(p)$ is the dual point corresponding to p

• A key result about Bregman divergences is that, given a random vector, the mean vector minimizes the expected Bregman divergence from the random vector. This result generalizes the textbook result that the mean of a set minimizes total squared error to elements in the set. This result was proved for the vector case by (Banerjee et al. 2005), and extended to the case of functions/distributions by (Frigyik et al. 2008). This result is important because it further justifies using a mean as a representative of a random set, particularly in Bayesian estimation.

Examples

• Squared Euclidean distance $D_F^y(x,y)=\Vert x-y{\Vert }^2$ is the canonical example of a Bregman distance, generated by the convex function $F(x)=\Vert x{\Vert }^2$

• The squared Mahalanobis distance, $D_F^y(x,y)=\frac{1}{2}(x-y{)}^{T}Q(x-y)$ which is generated by the convex function $F(x)=\frac{1}{2}{x}^{T}Qx$. This can be thought of as a generalization of the above squared Euclidean distance.

• The generalized Kullback–Leibler divergence

$D_F^q(p,q)=\sum p(i)\log \frac{p(i)}{q(i)}-\sum p(i)+\sum q(i)$

is generated by the convex function

$F(p)=\sum _{i}p(i)\log p(i)-\sum p(i)$

• The Itakura–Saito distance,

$D_F^q(p,q)=\sum _i(\frac{p(i)}{q(i)}-\log \frac{p(i)}{q(i)}-1)$

is generated by the convex function

$F(p)=-\sum \log p(i)$

Generalizing projective duality

A key tool in computational geometry is the idea of projective duality, which maps points to hyperplanes and vice versa, while preserving incidence and above-below relationships. There are numerous analytical forms of the projective dual: one common form maps the point $p=(p_1,\dots p_d)$ to the hyperplane $x_{d+1}=\sum _1^{d}2p_i{x}_i$. This mapping can be interpreted (identifying the hyperplane with its normal) as the convex conjugate mapping that takes the point p to its dual point $p^{*}=\mathrm{\nabla }F(p)$, where F defines the d-dimensional paraboloid $x_{d+1}=\sum x_i^2$.

If we now replace the paraboloid by an arbitrary convex function, we obtain a different dual mapping that retains the incidence and above-below properties of the standard projective dual. This implies that natural dual concepts in computational geometry like Voronoi diagrams and Delaunay triangulations retain their meaning in distance spaces defined by an arbitrary Bregman divergence. Thus, algorithms from "normal" geometry extend directly to these spaces (Boissonnat, Nielsen and Nock, 2010)

Matrix Bregman divergences, functional Bregman divergences and the submodular Bregman divergences

Bregman divergences can also be defined between matrices, between functions, and between measures (distributions). Bregman divergences between matrices include the Stein's loss and von Neumann entropy. Bregman divergences between functions include total squared error, relative entropy, and squared bias; see the references by Frigyik et al. below for definitions and properties. Similarly Bregman divergences have also been defined over sets, through a submodular set function which is known as the discrete analog of a convex function. The submodular Bregman divergences subsume a number of discrete distance measures, like the Hamming distance, precision and recall, mutual information and some other set based distance measures (see Iyer & Bilmes, 2012) for more details and properties of the submodular Bregman.)

For a list of common matrix Bregman divergences, see Table 15.1 in.

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
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• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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External links

• Bregman divergence interactive applet
• Bregman Voronoi diagram applet
• Exact Smallest Enclosing Bregman Ball applet
• Approximating smallest enclosing Bregman ball applet
• Sided and Symmetrized Bregman centroids