Siddharth Jain | California Institute of Technology (original) (raw)

Papers by Siddharth Jain

The majority of the human genome consists of repeated sequences. An important type of repeats com... more The majority of the human genome consists of repeated sequences. An important type of repeats common in the human genome are tandem repeats, where identical copies appear next to each other. For example, in the sequence AGT CT GT GC, T GT G is a tandem repeat, namely, generated from AGT CT GC by a tandem duplication of length 2. In this work, we investigate the possibility of generating a large number of sequences from a small initial string (called the seed) by tandem duplications of bounded length. Our results include exact capacity values for certain tandem duplication string systems with alphabet sizes 2, 3, and 4. In addition, motivated by the role of DNA sequences in expressing proteins via RNA and the genetic code, we define the notion of the expressiveness of a tandem duplication system, as the feasibility of expressing arbitrary substrings. We then completely characterize the expressiveness of tandem duplication systems for general alphabet sizes and duplication lengths. Noticing that a system with capacity = 1 is expressive, we prove that for an alphabet size ≥ 4, the capacity is strictly smaller than 1, independent of the seed and the duplication lengths. The proof of this limit on the capacity (note that the genomic alphabet size is 4), is related to an interesting result by Axel Thue from 1906 which states that there exist arbitrary length sequences with no tandem repeats (square-free) for alphabet size ≥ 3. Finally, our results illustrate that duplication lengths play a more significant role than the seed in generating a large number of sequences for these systems.

The Sliding Window Lempel-Ziv (SWLZ) algorithm that makes use of recurrence times and match lengt... more The Sliding Window Lempel-Ziv (SWLZ) algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature. In this paper, we undertake a finer study of these quantities under two different scenarios, i) zero entropy sources that are characterized by strong long-term memory, and ii) the processes with weak memory as described through various mixing conditions. For zero entropy sources, a general statement on match length is obtained. It is used in the proof of almost sure optimality of Fixed Shift Variant of Lempel-Ziv (FSLZ) and SWLZ algorithms given in literature. Through an example of stationary and ergodic processes generated by an irrational rotation we establish that for a window of size nw, a compression ratio given by O( log nw nw a ) where a depends on nw and approaches 1 as nw → ∞, is obtained under the application of FSLZ and SWLZ algorithms. Also, we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy.

Cardiac diseases, like those related to abnormal heart rate activity, have an enormous economic a... more Cardiac diseases, like those related to abnormal heart rate activity, have an enormous economic and psychological impact worldwide. The approaches used to control the behavior of modern pacemakers ignore the fractal nature of heart rate activity. The purpose of this paper is to present a Cyber Physical System approach towards pacemaker design which exploits precisely the fractal properties of heart rate activity in order to design the pacemaker controller.

Abstract—Beam forming is central to any wireless system. In different channel and signal scenario... more Abstract—Beam forming is central to any wireless system. In different channel and signal scenarios one can use MMSE or MSINR criteria for finding an optimal beamformer. It is well known in literature that if the channel is unknown then the solutions given by two criteria may not be the same, so we study the beam forming optimization problem with SINR-MSE tradeoffs which means given a minimum SINR threshold the problem focuses on finding the minimum MSE.

ABSTRACT The objective of this research is to understand the application of Genetic Algorithm in ... more ABSTRACT The objective of this research is to understand the application of Genetic Algorithm in the optimisation of design of CRLH Transmission Line (TL) Metamaterials. Metamaterials are a class of materials which exhibit properties not found in naturally occurring materials. Left Handed Metamaterials have negative ε (permittivity) and µ (permeability). The basic unit of a metamaterial is known as a unit cell. LHMs can be realised by a CRLH TL.

We extend the study by Ornstein and Weiss on the asymptotic behavior of the normalized version of... more We extend the study by Ornstein and Weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a certain class of mixing processes. Further, an estimator for entropy based on recurrence times is proposed for which large deviation behavior is proved for stationary and ergodic sources satisfying similar mixing conditions.

The Sliding Window Lempel-Ziv (SWLZ) algorithm has been studied from various perspectives in info... more The Sliding Window Lempel-Ziv (SWLZ) algorithm has been studied from various perspectives in information theory literature. In this paper, we provide a general law which defines the asymptotics of match length for stationary and ergodic zero entropy processes. Moreover, we use this law to choose the match length LoL_oLo in the almost sure optimality proof of Fixed Shift Variant of Lempel-Ziv (FSLZ) and SWLZ algorithms given in literature. First, through an example of stationary and ergodic processes generated by irrational rotation we establish that for a window size of nwn_wnw a compression ratio given by O(fraclognwnwa)O(\frac{\log n_w}{{n_w}^a})O(fraclognwnwa) where aaa is arbitrarily close to 1 and 0<a<10 < a < 10<a<1, is obtained under the application of FSLZ and SWLZ algorithms. Further, we give a general expression for the compression ratio for a class of stationary and totally ergodic processes with zero entropy.

Cardiac diseases, like those related to abnormal heart rate activity, have an enormous economic a... more Cardiac diseases, like those related to abnormal heart rate activity, have an enormous economic and psychological impact worldwide. The approaches used to control the behavior of modern pacemakers ignore the fractal nature of heart rate activity. The purpose of this paper is to present a Cyber Physical System approach towards pacemaker design which exploits precisely the fractal properties of heart rate activity in order to design the pacemaker controller. Towards the end, we solve a finite horizon optimal control problem based on the heart beat time series and show that this control problem can be converted into a system of linear equations. We also compare and contrast the performance of the fractal optimal control problem under six different cost functions. Finally, to get an idea of hardware complexity, we implement the fractal optimal controller on a Virtex4 FPGA and report some preliminary results in terms of area overhead.

Reducing energy consumption in multi-processor systems-on-chip (MPSoCs) where communication happe... more Reducing energy consumption in multi-processor systems-on-chip (MPSoCs) where communication happens via the network-on-chip (NoC) approach calls for multiple voltage/frequency island (VFI)-based designs. In turn, such multi-VFI architectures need efficient, robust, and accurate run-time control mechanisms that can exploit the workload characteristics in order to save power. Despite being tractable, the linear control models for power management cannot capture some important workload characteristics (e.g., fractality, non-stationarity) observed in heterogeneous NoCs, if ignored, such characteristics lead to inefficient communication and resources allocation, as well as high power dissipation in MPSoCs. To mitigate such limitations, we propose a new paradigm shift from power optimization based on linear models to control approaches based on fractal-state equations. As such, our approach is the first to propose a controller for fractal workloads with precise constraints on state and control variables and specific time bounds. Our results show that significant power savings (about 70%) can be achieved at run-time while running a variety of benchmark applications.

Managing cardiac disease and abnormal heart rate variability remain challenging problems with an ... more Managing cardiac disease and abnormal heart rate variability remain challenging problems with an enormous economic and psychological impact worldwide. Consequently, the purpose of this paper is to introduce a fractal approach to pacemaker design based on the constrained finite horizon optimal control problem. This is achieved by modeling the heart rate dynamics via fractional differential equations. Also, by using calculus of variations, we show that the constrained finite horizon optimal control problem can be reduced to solving a linear system. Finally, we discuss the hardware complexity involved in the practical implementation of fractal controllers.

Talks by Siddharth Jain

The majority of the human genome consists of repeated sequences. An important type of repeats com... more The majority of the human genome consists of repeated sequences. An important type of repeats common in the human genome are tandem repeats, where identical copies appear next to each other. For example, in the sequence AGT CT GT GC, T GT G is a tandem repeat, namely, generated from AGT CT GC by a tandem duplication of length 2. In this work, we investigate the possibility of generating a large number of sequences from a small initial string (called the seed) by tandem duplications of bounded length. Our results include exact capacity values for certain tandem duplication string systems with alphabet sizes 2, 3, and 4. In addition, motivated by the role of DNA sequences in expressing proteins via RNA and the genetic code, we define the notion of the expressiveness of a tandem duplication system, as the feasibility of expressing arbitrary substrings. We then completely characterize the expressiveness of tandem duplication systems for general alphabet sizes and duplication lengths. Noticing that a system with capacity = 1 is expressive, we prove that for an alphabet size ≥ 4, the capacity is strictly smaller than 1, independent of the seed and the duplication lengths. The proof of this limit on the capacity (note that the genomic alphabet size is 4), is related to an interesting result by Axel Thue from 1906 which states that there exist arbitrary length sequences with no tandem repeats (square-free) for alphabet size ≥ 3. Finally, our results illustrate that duplication lengths play a more significant role than the seed in generating a large number of sequences for these systems.

The Sliding Window Lempel-Ziv (SWLZ) algorithm that makes use of recurrence times and match lengt... more The Sliding Window Lempel-Ziv (SWLZ) algorithm that makes use of recurrence times and match lengths has been studied from various perspectives in information theory literature. In this paper, we undertake a finer study of these quantities under two different scenarios, i) zero entropy sources that are characterized by strong long-term memory, and ii) the processes with weak memory as described through various mixing conditions. For zero entropy sources, a general statement on match length is obtained. It is used in the proof of almost sure optimality of Fixed Shift Variant of Lempel-Ziv (FSLZ) and SWLZ algorithms given in literature. Through an example of stationary and ergodic processes generated by an irrational rotation we establish that for a window of size nw, a compression ratio given by O( log nw nw a ) where a depends on nw and approaches 1 as nw → ∞, is obtained under the application of FSLZ and SWLZ algorithms. Also, we give a general expression for the compression ratio for a class of stationary and ergodic processes with zero entropy.

Cardiac diseases, like those related to abnormal heart rate activity, have an enormous economic a... more Cardiac diseases, like those related to abnormal heart rate activity, have an enormous economic and psychological impact worldwide. The approaches used to control the behavior of modern pacemakers ignore the fractal nature of heart rate activity. The purpose of this paper is to present a Cyber Physical System approach towards pacemaker design which exploits precisely the fractal properties of heart rate activity in order to design the pacemaker controller.

Abstract—Beam forming is central to any wireless system. In different channel and signal scenario... more Abstract—Beam forming is central to any wireless system. In different channel and signal scenarios one can use MMSE or MSINR criteria for finding an optimal beamformer. It is well known in literature that if the channel is unknown then the solutions given by two criteria may not be the same, so we study the beam forming optimization problem with SINR-MSE tradeoffs which means given a minimum SINR threshold the problem focuses on finding the minimum MSE.

ABSTRACT The objective of this research is to understand the application of Genetic Algorithm in ... more ABSTRACT The objective of this research is to understand the application of Genetic Algorithm in the optimisation of design of CRLH Transmission Line (TL) Metamaterials. Metamaterials are a class of materials which exhibit properties not found in naturally occurring materials. Left Handed Metamaterials have negative ε (permittivity) and µ (permeability). The basic unit of a metamaterial is known as a unit cell. LHMs can be realised by a CRLH TL.

We extend the study by Ornstein and Weiss on the asymptotic behavior of the normalized version of... more We extend the study by Ornstein and Weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a certain class of mixing processes. Further, an estimator for entropy based on recurrence times is proposed for which large deviation behavior is proved for stationary and ergodic sources satisfying similar mixing conditions.

The Sliding Window Lempel-Ziv (SWLZ) algorithm has been studied from various perspectives in info... more The Sliding Window Lempel-Ziv (SWLZ) algorithm has been studied from various perspectives in information theory literature. In this paper, we provide a general law which defines the asymptotics of match length for stationary and ergodic zero entropy processes. Moreover, we use this law to choose the match length LoL_oLo in the almost sure optimality proof of Fixed Shift Variant of Lempel-Ziv (FSLZ) and SWLZ algorithms given in literature. First, through an example of stationary and ergodic processes generated by irrational rotation we establish that for a window size of nwn_wnw a compression ratio given by O(fraclognwnwa)O(\frac{\log n_w}{{n_w}^a})O(fraclognwnwa) where aaa is arbitrarily close to 1 and 0<a<10 < a < 10<a<1, is obtained under the application of FSLZ and SWLZ algorithms. Further, we give a general expression for the compression ratio for a class of stationary and totally ergodic processes with zero entropy.

Cardiac diseases, like those related to abnormal heart rate activity, have an enormous economic a... more Cardiac diseases, like those related to abnormal heart rate activity, have an enormous economic and psychological impact worldwide. The approaches used to control the behavior of modern pacemakers ignore the fractal nature of heart rate activity. The purpose of this paper is to present a Cyber Physical System approach towards pacemaker design which exploits precisely the fractal properties of heart rate activity in order to design the pacemaker controller. Towards the end, we solve a finite horizon optimal control problem based on the heart beat time series and show that this control problem can be converted into a system of linear equations. We also compare and contrast the performance of the fractal optimal control problem under six different cost functions. Finally, to get an idea of hardware complexity, we implement the fractal optimal controller on a Virtex4 FPGA and report some preliminary results in terms of area overhead.

Reducing energy consumption in multi-processor systems-on-chip (MPSoCs) where communication happe... more Reducing energy consumption in multi-processor systems-on-chip (MPSoCs) where communication happens via the network-on-chip (NoC) approach calls for multiple voltage/frequency island (VFI)-based designs. In turn, such multi-VFI architectures need efficient, robust, and accurate run-time control mechanisms that can exploit the workload characteristics in order to save power. Despite being tractable, the linear control models for power management cannot capture some important workload characteristics (e.g., fractality, non-stationarity) observed in heterogeneous NoCs, if ignored, such characteristics lead to inefficient communication and resources allocation, as well as high power dissipation in MPSoCs. To mitigate such limitations, we propose a new paradigm shift from power optimization based on linear models to control approaches based on fractal-state equations. As such, our approach is the first to propose a controller for fractal workloads with precise constraints on state and control variables and specific time bounds. Our results show that significant power savings (about 70%) can be achieved at run-time while running a variety of benchmark applications.

Managing cardiac disease and abnormal heart rate variability remain challenging problems with an ... more Managing cardiac disease and abnormal heart rate variability remain challenging problems with an enormous economic and psychological impact worldwide. Consequently, the purpose of this paper is to introduce a fractal approach to pacemaker design based on the constrained finite horizon optimal control problem. This is achieved by modeling the heart rate dynamics via fractional differential equations. Also, by using calculus of variations, we show that the constrained finite horizon optimal control problem can be reduced to solving a linear system. Finally, we discuss the hardware complexity involved in the practical implementation of fractal controllers.