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Jian Tan

Publications and source records attributed to Jian Tan.

At least 37 records · Page 2Linked to original sources
arXiv2021

Real-Variable Theory of Local Variable Hardy Spaces

In this paper, we give a complete real-variable theory of local variable Hardy spaces. First, we present various real-variable characterizations in terms of several local maximal functions. Next, the new atomic and the finite atomic decomposition for the local variable Hardy spaces are established. As an application, we also introduce the local variable Campanato space which is showed to be the dual space of the local variable Hardy spaces. Analogous to the homogeneous case, some equivalent definitions of the dual of local variable Hardy spaces are also considered. Finally, we show that the boundedness of inhomogeneous Calderón-Zygmund singular integrals and local fractional integrals on local variable Hardy spaces and their duals.

math.CA↗
arXiv2020

Local Differential Privacy for Bayesian Optimization

Motivated by the increasing concern about privacy in nowadays data-intensive online learning systems, we consider a black-box optimization in the nonparametric Gaussian process setting with local differential privacy (LDP) guarantee. Specifically, the rewards from each user are further corrupted to protect privacy and the learner only has access to the corrupted rewards to minimize the regret. We first derive the regret lower bounds for any LDP mechanism and any learning algorithm. Then, we present three almost optimal algorithms based on the GP-UCB framework and Laplace DP mechanism. In this process, we also propose a new Bayesian optimization (BO) method (called MoMA-GP-UCB) based on median-of-means techniques and kernel approximations, which complements previous BO algorithms for heavy-tailed payoffs with a reduced complexity. Further, empirical comparisons of different algorithms on both synthetic and real-world datasets highlight the superior performance of MoMA-GP-UCB in both private and non-private scenarios.

cs.LG↗
arXiv2020

Learning Latent Features with Pairwise Penalties in Low-Rank Matrix Completion

Low-rank matrix completion has achieved great success in many real-world data applications. A matrix factorization model that learns latent features is usually employed and, to improve prediction performance, the similarities between latent variables can be exploited by pairwise learning using the graph regularized matrix factorization (GRMF) method. However, existing GRMF approaches often use the squared loss to measure the pairwise differences, which may be overly influenced by dissimilar pairs and lead to inferior prediction. To fully empower pairwise learning for matrix completion, we propose a general optimization framework that allows a rich class of (non-)convex pairwise penalty functions. A new and efficient algorithm is developed to solve the proposed optimization problem, with a theoretical convergence guarantee under mild assumptions. In an important situation where the latent variables form a small number of subgroups, its statistical guarantee is also fully considered. In particular, we theoretically characterize the performance of the complexity-regularized maximum likelihood estimator, as a special case of our framework, which is shown to have smaller errors when compared to the standard matrix completion framework without pairwise penalties. We conduct extensive experiments on both synthetic and real datasets to demonstrate the superior performance of this general framework.

stat.ML↗
arXiv2019

RobustTrend: A Huber Loss with a Combined First and Second Order Difference Regularization for Time Series Trend Filtering

Extracting the underlying trend signal is a crucial step to facilitate time series analysis like forecasting and anomaly detection. Besides noise signal, time series can contain not only outliers but also abrupt trend changes in real-world scenarios. To deal with these challenges, we propose a robust trend filtering algorithm based on robust statistics and sparse learning. Specifically, we adopt the Huber loss to suppress outliers, and utilize a combination of the first order and second order difference on the trend component as regularization to capture both slow and abrupt trend changes. Furthermore, an efficient method is designed to solve the proposed robust trend filtering based on majorization minimization (MM) and alternative direction method of multipliers (ADMM). We compared our proposed robust trend filter with other nine state-of-the-art trend filtering algorithms on both synthetic and real-world datasets. The experiments demonstrate that our algorithm outperforms existing methods.

cs.LG↗
arXiv2019

Discrete para-product operators on variable Hardy spaces

Let $p(\cdot):\mathbb R^n\rightarrow(0,\infty)$ be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of para-product operators $π_b$ on variable Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$, where $b\in BMO(\mathbb R^n)$. As an application, we show that non-convolution type Calderón-Zygmund operators $T$ are bounded on $H^{p(\cdot)}(\mathbb R^n)$ if and only if $T^\ast1=0$, where $\frac{n}{n+ε}<\mbox{essinf}_{x\in\mathbb R^n} p\le \mbox{esssup}_{x\in\mathbb R^n} p\le 1$, $ε$ is the regular exponent of kernel of $T$. Our approach relies on the discrete version of Calderón's reproducing formula, discrete Littlewood-Paley-Stein theory and almost orthogonal estimates. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

math.CA↗
arXiv2019

Carleson measure spaces with variable exponents and their applications

In this paper, we introduce the Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$. By using discrete Littlewood$-$Paley$-$Stein analysis as well as Frazier and Jawerth's $φ-$transform in the variable exponent settings, we show that the dual space of the variable Hardy space $H^{p(\cdot)}$ is $CMO^{p(\cdot)}$. As applications, we obtain that Carleson measure spaces with variable exponents $CMO^{p(\cdot)}$, Campanato space with variable exponent $\mathfrak{L}_{q,p(\cdot),d}$ and Hölder-Zygmund spaces with variable exponents $\mathcal {\dot{H}}_d^{p(\cdot)}$ coincide as sets and the corresponding norms are equivalent. Via using an argument of weak density property, we also prove the boundedness of Calderón-Zygmund singular integral operator acting on $CMO^{p(\cdot)}$.

math.CA↗
arXiv2018

Heavy-traffic Delay Optimality in Pull-based Load Balancing Systems: Necessary and Sufficient Conditions

In this paper, we consider a load balancing system under a general pull-based policy. In particular, each arrival is randomly dispatched to one of the servers whose queue lengths are below a threshold, if there are any; otherwise, this arrival is randomly dispatched to one of the entire set of servers. We are interested in the fundamental relationship between the threshold and the delay performance of the system in heavy traffic. To this end, we first establish the following necessary condition to guarantee heavy-traffic delay optimality: the threshold will grow to infinity as the exogenous arrival rate approaches the boundary of the capacity region (i.e., the load intensity approaches one) but the growth rate should be slower than a polynomial function of the mean number of tasks in the system. As a special case of this result, we directly show that the delay performance of the popular pull-based policy Join-Idle-Queue (JIQ) lies strictly between that of any heavy-traffic delay optimal policy and that of random routing. We further show that a sufficient condition for heavy-traffic delay optimality is that the threshold grows logarithmically with the mean number of tasks in the system. This result directly resolves a generalized version of the conjecture by Kelly and Laws.

cs.PF↗
arXiv2018

Flexible Load Balancing with Multi-dimensional State-space Collapse: Throughput and Heavy-traffic Delay Optimality

Heavy traffic analysis for load balancing policies has relied heavily on the condition of state-space collapse onto a single-dimensional line in previous works. In this paper, via Lyapunov-drift analysis, we rigorously prove that even under a multi-dimensional state-space collapse, steady-state heavy-traffic delay optimality can still be achieved for a general load balancing system. This result directly implies that achieving steady-state heavy-traffic delay optimality simply requires that no server is idling while others are busy at heavy loads, thus complementing and extending the result obtained by diffusion approximations. Further, we explore the greater flexibility provided by allowing a multi-dimensional state-space collapse in designing new load balancing policies that are both throughput optimal and heavy-traffic delay optimal in steady state. This is achieved by overcoming various technical challenges, and the methods used in this paper could be of independent interest.

cs.DC↗
arXiv2018

Fork and Join Queueing Networks with Heavy Tails: Scaling Dimension and Throughput Limit

Parallel and distributed computing systems are foundational to the success of cloud computing and big data analytics. These systems process computational workflows in a way that can be mathematically modeled by a fork-and-join queueing network with blocking (FJQN/B). While engineering solutions have long been made to build and scale such systems, it is challenging to rigorously characterize their throughput performance at scale theoretically. What further complicates the study is the presence of heavy-tailed delays that have been widely documented therein. To this end, we introduce two fundamental concepts for networks of arbitrary topology (scaling dimension and extended metric dimension) and utilize an infinite sequence of growing FJQN/Bs to study the throughput limit. The throughput is said to be scalable if the throughput limit infimum of the sequence is strictly positive as the network size grows to infinity. We investigate throughput scalability by focusing on heavy-tailed service times that are regularly varying (with index $α>1$) and featuring the network topology described by the two aforementioned dimensions. In particular, we show that an infinite sequence of FJQN/Bs is throughput scalable if the extended metric dimension $<α-1$ and only if the scaling dimension $\leα-1$. These theoretical results provide new insights on the scalability of a rich class of FJQN/Bs with various structures, including tandem, lattice, hexagon, pyramid, tree, and fractals.

cs.DC↗
arXiv2018

Asymptotic Miss Ratio of LRU Caching with Consistent Hashing

To efficiently scale data caching infrastructure to support emerging big data applications, many caching systems rely on consistent hashing to group a large number of servers to form a cooperative cluster. These servers are organized together according to a random hash function. They jointly provide a unified but distributed hash table to serve swift and voluminous data item requests. Different from the single least-recently-used (LRU) server that has already been extensively studied, theoretically characterizing a cluster that consists of multiple LRU servers remains yet to be explored. These servers are not simply added together; the random hashing complicates the behavior. To this end, we derive the asymptotic miss ratio of data item requests on a LRU cluster with consistent hashing. We show that these individual cache spaces on different servers can be effectively viewed as if they could be pooled together to form a single virtual LRU cache space parametrized by an appropriate cache size. This equivalence can be established rigorously under the condition that the cache sizes of the individual servers are large. For typical data caching systems this condition is common. Our theoretical framework provides a convenient abstraction that can directly apply the results from the simpler single LRU cache to the more complex LRU cluster with consistent hashing.

cs.PF↗
arXiv2017

Designing Low-Complexity Heavy-Traffic Delay-Optimal Load Balancing Schemes: Theory to Algorithms

We establish a unified analytical framework for load balancing systems, which allows us to construct a general class $Π$ of policies that are both throughput optimal and heavy-traffic delay optimal. This general class $Π$ includes as special cases popular policies such as join-shortest-queue and power-of-$d$, but not the join-idle-queue (JIQ) policy. In fact, we show that JIQ, which is not in $Π$, is actually not heavy-traffic delay optimal. Owing to the significant flexibility offered by class $Π$, we are able to design a new policy called join-below-threshold (JBT-d), which maintains the simplicity of pull-based policies such as JIQ, but updates its threshold dynamically. We prove that JBT-$d$ belongs to the class $Π$ when the threshold is picked appropriately and thus it is heavy-traffic delay optimal. Extensive simulations show that the new policy not only has a low complexity in message rates, but also achieves excellent delay performance, comparable to the optimal join-shortest-queue in various system settings.

cs.DC↗
arXiv2017

On Resource Pooling and Separation for LRU Caching

Caching systems using the Least Recently Used (LRU) principle have now become ubiquitous. A fundamental question for these systems is whether the cache space should be pooled together or divided to serve multiple flows of data item requests in order to minimize the miss probabilities. In this paper, we show that there is no straight yes or no answer to this question, depending on complex combinations of critical factors, including, e.g., request rates, overlapped data items across different request flows, data item popularities and their sizes. Specifically, we characterize the asymptotic miss probabilities for multiple competing request flows under resource pooling and separation for LRU caching when the cache size is large. Analytically, we show that it is asymptotically optimal to jointly serve multiple flows if their data item sizes and popularity distributions are similar and their arrival rates do not differ significantly; the self-organizing property of LRU caching automatically optimizes the resource allocation among them asymptotically. Otherwise, separating these flows could be better, e.g., when data sizes vary significantly. We also quantify critical points beyond which resource pooling is better than separation for each of the flows when the overlapped data items exceed certain levels. Technically, we generalize existing results on the asymptotic miss probability of LRU caching for a broad class of heavy-tailed distributions and extend them to multiple competing flows with varying data item sizes, which also validates the Che approximation under certain conditions. These results provide new insights on improving the performance of caching systems.

cs.PF↗
arXiv2015

Yeast caspase 1 suppresses the burst of reactive oxygen species and maintains mitochondrial stability in Saccharomyces cerevisiae

Caspases are a family of cysteine proteases that play essential roles during apoptosis, and we presume some of them may also protect the cell from oxidative stress. We found that the absence of yeast caspase 1(Yca1)in Saccharomyces cerevisiae leads to a more intense burst of mitochondrial reactive oxygen species (ROS) In addition, compared to wild type yeast cells, the ability of yca1 mutant cells to maintain mitochondrial activity is significantly reduced after either oxidative stress treatment or aging. During mitochondrial ROS burst, deletion of the yca1 gene delayed structural damage of a green fluorescent protein (GFP) reporter bound in the inner mitochondrial membrane. This work implies that yeast caspase 1 is closely connected to the oxidative stress response. We speculate that Yca1 can discriminate proteins damaged by oxidation and accelerate their hydrolysis to attenuate the ROS burst.

q-bio.CB↗
arXiv2012

Delay Asymptotics with Retransmissions and Incremental Redundancy Codes over Erasure Channels

Recent studies have shown that retransmissions can cause heavy-tailed transmission delays even when packet sizes are light-tailed. Moreover, the impact of heavy-tailed delays persists even when packets size are upper bounded. The key question we study in this paper is how the use of coding techniques to transmit information, together with different system configurations, would affect the distribution of delay. To investigate this problem, we model the underlying channel as a Markov modulated binary erasure channel, where transmitted bits are either received successfully or erased. Erasure codes are used to encode information prior to transmission, which ensures that a fixed fraction of the bits in the codeword can lead to successful decoding. We use incremental redundancy codes, where the codeword is divided into codeword trunks and these trunks are transmitted one at a time to provide incremental redundancies to the receiver until the information is recovered. We characterize the distribution of delay under two different scenarios: (I) Decoder uses memory to cache all previously successfully received bits. (II) Decoder does not use memory, where received bits are discarded if the corresponding information cannot be decoded. In both cases, we consider codeword length with infinite and finite support. From a theoretical perspective, our results provide a benchmark to quantify the tradeoff between system complexity and the distribution of delay.

cs.IT↗
arXiv2011

Secrecy Outage Capacity of Fading Channels

This paper considers point to point secure communication over flat fading channels under an outage constraint. More specifically, we extend the definition of outage capacity to account for the secrecy constraint and obtain sharp characterizations of the corresponding fundamental limits under two different assumptions on the transmitter CSI (Channel state information). First, we find the outage secrecy capacity assuming that the transmitter has perfect knowledge of the legitimate and eavesdropper channel gains. In this scenario, the capacity achieving scheme relies on opportunistically exchanging private keys between the legitimate nodes. These keys are stored in a key buffer and later used to secure delay sensitive data using the Vernam's one time pad technique. We then extend our results to the more practical scenario where the transmitter is assumed to know only the legitimate channel gain. Here, our achievability arguments rely on privacy amplification techniques to generate secret key bits. In the two cases, we also characterize the optimal power control policies which, interestingly, turn out to be a judicious combination of channel inversion and the optimal ergodic strategy. Finally, we analyze the effect of key buffer overflow on the overall outage probability.

cs.IT↗
arXiv2009

Stability of Finite Population ALOHA with Variable Packets

ALOHA is one of the most basic Medium Access Control (MAC) protocols and represents a foundation for other more sophisticated distributed and asynchronous MAC protocols, e.g., CSMA. In this paper, unlike in the traditional work that focused on mean value analysis, we study the distributional properties of packet transmission delays over an ALOHA channel. We discover a new phenomenon showing that a basic finite population ALOHA model with variable size (exponential) packets is characterized by power law transmission delays, possibly even resulting in zero throughput. These results are in contrast to the classical work that shows exponential delays and positive throughput for finite population ALOHA with fixed packets. Furthermore, we characterize a new stability condition that is entirely derived from the tail behavior of the packet and backoff distributions that may not be determined by mean values. The power law effects and the possible instability might be diminished, or perhaps eliminated, by reducing the variability of packets. However, we show that even a slotted (synchronized) ALOHA with packets of constant size can exhibit power law delays when the number of active users is random. From an engineering perspective, our results imply that the variability of packet sizes and number of active users need to be taken into consideration when designing robust MAC protocols, especially for ad-hoc/sensor networks where other factors, such as link failures and mobility, might further compound the problem.

cs.PF↗
arXiv2008

Modulated Branching Processes, Origins of Power Laws and Queueing Duality

Power law distributions have been repeatedly observed in a wide variety of socioeconomic, biological and technological areas. In many of the observations, e.g., city populations and sizes of living organisms, the objects of interest evolve due to the replication of their many independent components, e.g., births-deaths of individuals and replications of cells. Furthermore, the rates of the replication are often controlled by exogenous parameters causing periods of expansion and contraction, e.g., baby booms and busts, economic booms and recessions, etc. In addition, the sizes of these objects often have reflective lower boundaries, e.g., cities do not fall bellow a certain size, low income individuals are subsidized by the government, companies are protected by bankruptcy laws, etc. Hence, it is natural to propose reflected modulated branching processes as generic models for many of the preceding observations. Indeed, our main results show that the proposed mathematical models result in power law distributions under quite general polynomial Gartner-Ellis conditions, the generality of which could explain the ubiquitous nature of power law distributions. In addition, on a logarithmic scale, we establish an asymptotic equivalence between the reflected branching processes and the corresponding multiplicative ones. The latter, as recognized by Goldie (1991), is known to be dual to queueing/additive processes. We emphasize this duality further in the generality of stationary and ergodic processes.

math.PR↗