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Let Φ : R N R M denote a sensing matrix and Δ : R M R N denote a recovery algorithm. We say that the pair ( Φ , Δ ) is C -stable if for any x Σ K and any e R M we have that

Δ Φ x + e - x 2 C e .

This definition simply says that if we add a small amount of noise to the measurements, then the impact of this on the recovered signal should not be arbitrarily large. [link] below demonstrates that the existence of any decoding algorithm (potentially impractical) that can stably recover from noisy measurements requires that Φ satisfy the lower bound of [link] with a constant determined by C .

  If the pair ( Φ , Δ ) is C -stable, then

1 C x 2 Φ x 2

for all x Σ 2 K .

Pick any x , z Σ K . Define

e x = Φ ( z - x ) 2 and e z = Φ ( x - z ) 2 ,

and note that

Φ x + e x = Φ z + e z = Φ ( x + z ) 2 .

Let x ^ = Δ ( Φ x + e x ) = Δ ( Φ z + e z ) . From the triangle inequality and the definition of C -stability, we have that

x - z 2 = x - x ^ + x ^ - z 2 x - x ^ 2 + x ^ - z 2 C e x + C e z 2 = C Φ x - Φ z 2 .

Since this holds for any x , z Σ K , the result follows.

Note that as C 1 , we have that Φ must satisfy the lower bound of [link] with δ K = 1 - 1 / C 2 0 . Thus, if we desire to reduce the impact of noise in our recovered signal then we must adjust Φ so that it satisfies the lower bound of [link] with a tighter constant.

One might respond to this result by arguing that since the upper bound is not necessary, we can avoid redesigning Φ simply by rescaling Φ so that as long as Φ satisfies the RIP with δ 2 K < 1 , the rescaled version α Φ will satisfy [link] for any constant C . In settings where the size of the noise is independent of our choice of Φ , this is a valid point — by scaling Φ we are simply adjusting the gain on the “signal” part of our measurements, and if increasing this gain does not impact the noise, then we can achieve arbitrarily high signal-to-noise ratios, so that eventually the noise is negligible compared to the signal.

However, in practice we will typically not be able to rescale Φ to be arbitrarily large. Moreover, in many practical settings the noise is not independent of Φ . For example, suppose that the noise vector e represents quantization noise produced by a finite dynamic range quantizer with B bits. Suppose the measurements lie in the interval [ - T , T ] , and we have adjusted the quantizer to capture this range. If we rescale Φ by α , then the measurements now lie between [ - α T , α T ] , and we must scale the dynamic range of our quantizer by α . In this case the resulting quantization error is simply α e , and we have achieved no reduction in the reconstruction error.

Measurement bounds

We can also consider how many measurements are necessary to achieve the RIP. If we ignore the impact of δ and focus only on the dimensions of the problem ( N , M , and K ) then we can establish a simple lower bound. We first provide a preliminary lemma that we will need in the proof of the main theorem.

Let K and N satisfying K < N / 2 be given. There exists a set X Σ K such that for any x X we have x 2 K and for any x , z X with x z ,

x - z 2 K / 2 ,


log | X | K 2 log N K .

We will begin by considering the set

U = x 0 , + 1 , - 1 N : x 0 = K .

By construction, x 2 2 = K for all x U . Thus if we construct X by picking elements from U then we automatically have x 2 K .

Next, observe that | U | = N K 2 K . Note also that x - z 0 x - z 2 2 , and thus if x - z 2 2 K / 2 then x - z 0 K / 2 . From this we observe that for any fixed x U ,

Questions & Answers

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oyebola Reply
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Alemogho Reply
ask the question you don't know
opportunity cost
the motive of money is to pay for goods or services. it's also for transaction purposes bet it ordinary or unforseen
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Edmond Reply
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and now the solving techinique is also changed. only theories or its matematical +statistical version is not sufficient so the economist had developed Econometrics which includes economic theory+mathematical economics+statistics, for solving economic problem. This method treats economics as a pure
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measurement of elasticity of demand
John Reply
Price elasticity of demand= Percentage change in quantity demanded/ Percentage change in price
For example, suppose that a 10 percent increase in the price of an ice-cream cone causes the amount of ice cream you buy to fall by 20 percent. We calculate your elasticity of demand as
Price elasticity of demand =20 percent /10 percent = 2
In this example, the elasticity is 2, reflecting that the change in the quantity demanded is proportionately twice as large as the change in the price.
what is economic
Samuel Reply
economics is a social science that studies the relationship of human behavior between ends and scare...
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it's significant in study of managing available resources to satisfy human needs and wants
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it is also said that it is the science of scarcity
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Economic helps to meet our needs and wants nd to make choices.
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Penda Reply
what is equilibrium price?
Stella Reply
happen when demand curve and supply curve meet
intersection point of demand and supply curve
the equilibrium price is that price at which the quantity demanded is equal to the quantity supplied... i.e the price at which what people want to buy is equal to what the sellers want to sell.
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the price which is prevailing in the market ,it is the price at which the consumer is getting maximum satisfaction and the producer is getting maximum profit.
at equilibrium price demand for commodity is equal to supply of commodity..
dear sareel, quantity demand means the quantity of commodity which is demanded at a particular price e,g , your demand is for 4 oranges when price of orange is 5,, on the other hand demand means the whole quantity price relationship, quantity demand is shown on a particular point on demand curve,
sarjeel sbb, reply please...
what is elasticity of demand
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Economics can be defined as a social sciences that study human behavior as a relationship between end and scarce mean which have alternative use
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economics is a subject matter that studies different economic activities as directed towards maximization of income at level of individual and maximization of social welfare at the level of society as a whole
economics studies us how to allocate scarce resources to full fill the maximum human wants as the resources have alternative uses..and r very scare to fullfill the human wants as the wants of humans are unlimited
demand is what a buyer is able and willing to buy at a specific price where by quantity demanded is amount of demand the buyer is able and willing to buy at a specific price.
thanks che..
how does commercial banks create credit
write models of Aggregate Supply?
abdi Reply
models of aggregate supply?
Then what's the law of elasticity?
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Enockz Reply
the amount of money in circulation increases and cause aggregate demand to rise. Thus, hyperinflation.
what's heteroscandacity
price mechanism refers to the manner which price of goods and services affect the demand and supply of goods and services
heteroscandacity: Is the circumstance where there is unequal variances across the observations.
what is price mechanisms?
Che Reply
price mechanism refers to the manner which price of goods and services affect the demand and supply of goods and services
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kabeer Reply
desire and ability to pay for a commodity is called demand.
i desire to become Prime Minister of India , is it demand ?please answer---
demand refers to the quantity demanded of a commodity that consumers are willing and able to buy at a particular price over a given period of time
or the willingness power of a commoditythat is backed up by the ability to pay
u r 100% right..
is ability to purchase a perticular goods at any price at a given time.
tanz Rather
ur desire to become a prime minister of lndia is nt demand because u have nt made any efforts or willingness, it is called mere wish or lets say mere desire
Thank you so much dear Abigail Mante for giving answer to my question..
U are welcome
it's ok,dear..
The term ceteris paribus implies that
Demand is the quantity demanded of a commodity at which a consumer is willing and able to purchase at various prices over a given period of time.
ceteris parbus explains the law of demand,which states that at a higher price quantity demanded of a commodity is low and at a lower price quantity demanded of a commodity is high all other things being equal.
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ceteris paribus is an economics terms used in situations where we considered other factors to remain unchanged
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please how does disposable income affect demand?
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the quantity of a commodity that a consumer ready willing and able to purchase at a particular price and in a particular period of time is known as demand
ceterusparibus means constant i,e,(other things do not change or other things remains constant)
right answer yawer saab
things like what@Rather
bilal sheikh why u has been written what
e,g demand is determined by various factors like price of commodity, uses of good, expected changes in near future, nature of commodity , but when we learn price elasticity of demand , we keep other determinants of demand constant(i,e , we do not expect any change in price of good in near future,etc
i think u understand the term ceterus paribus..
bcz I can not understand
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wht is other factors
income of consumer, taste,
Gold. Medlist in economics and. Lecturer of economics
hello..what's are the factors affecting cross elasticity of demand.? at least five factors
The time it takes to get a product to market. That time limit has a name call "Just in Time."
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when there is no change in quantity demanded due to change in price then elasticity of demand is inelastic
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yes. what is the main cause of Heteroscedasticity
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elasticity of demand means change in quantity demanded divided by change in any of its determinent ,e,g , price ed= % change in quantity demanded/ % change in price.
elasticity of demand is the degree of responsiveness of quantity demanded of a commodity to a change in price of the commodity in question ,price of other related commodity n the consumers income
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A single seller in the market.
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Eric Reply
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adedayo Reply
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Micheal Reply
a device agreed upon by a group of people to be used interchangeably for goods and/or services.
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in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
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Source:  OpenStax, An introduction to compressive sensing. OpenStax CNX. Apr 02, 2011 Download for free at http://legacy.cnx.org/content/col11133/1.5
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