Monday

Dynamical segmentation of single trials from population neural data
Biljana Petreska, M. Sahani, B. Yu, J. Cunningham, S. Ryu, K. Shenoy, Gopal Santhanam

A randomly switching piecewise-linear dynamical system model is constructed via discrete latent states. Given a state, the dynamics of spiking neurons are assumed to be linear. This model is fit to 105 simultaneously recorded neurons (Utah array) during a motor task. Number of states were chosen heuristically. This is an unsupervised method that automatically captures the structure of the dynamics. The results suggest that neurons tend to be in a linear dynamical state both when waiting for the go-cue, and during early movement, and goes through nonlinear dynamical transitions in between.

Inferring spike-timing-dependent plasticity from spike train data
Ian H. Stevenson, Konrad P. Kording

Different synapses have different form of STDP, and while spike train data are abundant, in vivo whole cell recordings are very difficult. Hence, learning the synaptic plasticity rule from just spike train observation is of great importance. This is one of my long-term goals as well. They fit a unidirectionally coupled GLM model with a binned weight modulation function as a function of timing to previous presynaptic spike. The results are promising for simulated models. I’d love to see it applied to a well controlled real data.

Active dendrites: adaptation to spike-based communication
Balázs B Ujfalussy, Máté Lengyel

In the presence of correlated presynaptic population activity, to compute a function of presynaptic voltage online from spikes, the neuron has to be nonlinear. In particular, this paper links it to the nonlinear summation property of the dendrite. In previous work by Pfister, J., Dayan, P., Lengyel, M. (2010), they explained the role of short-term plasticity (dynamical synapse model) as optimal predictor for presynaptic membrane potential for a single neuron. This work expands it to the population case.

From stochastic nonlinear integrate-and-fire to generalized linear models
Skander Mensi, Richard Naud, Wulfram Gersnter

This poster shows that given a stochastic (adaptive-exponential) leaky-integrate-and-fire-neuron model, it is possible to construct a nearly equivalent GLM model (as a form of spike response model (SRM) with escape noise). Sub-threshold dynamics is linearized to provide the linear filter (corresponding to impulse response) and the reset/refractoriness part of the history filter, while the spike-adaptation is captured as a slower time scale component of the history filter. Then the link function can be estimated through empirical observation that is close to being linear. (I was totally thrown off by the notation which was the probability of spiking given a membrane potential, not the marginal distribution of voltage distribution of the model.)

Gaussian process modulated renewal processes
Vinayak Rao, Yee Whye Teh

This is an extension of R. P. Adams, I. Murray and D. J.C. MacKay’s work which was on Poisson intensity estimation to hazard rate modulated renewal process. Basic ideas are similar; use a sigmoidal link function, and use point process thinning like procedure to exactly sample.

Tuesday

Learning in Hilbert vs. Banach spaces: A measure embedding viewpoint
Bharath K. Sriperumbudur, Kenji Fukumizu, Gert R. G. Lanckriet

Kernel embedding of probability distribution and induced divergence is an emerging direction of kernel methods. The divergence is related to Bayes risk of Parzen window classifier in particular, and this paper extends the results to Banach spaces. For a Banach space with a norm that is uniformly Fretchet differentiable, and uniformly convex, there is a semi-inner product inducing an reproducing kernel Banach space (RKBS) which has analogous properties to RKHS. They showed that kernel embedding is injective when the kernel is a Fourier transform of a signed measure (c.f. Bochner’s theorem requires a positive measure for positive definiteness). The resulting divergence is not computable, unless the semi-metric is of special form, and the convergence rate turns out to be at best same as the RKHS case.

Modelling genetic variations with fragmentation-coagulation processes
Yee Whye Teh, Charles Blundell, Lloyd T. Elliott

Similar to Chinese restaurant process (CRP) for clustering, a temporal evolution of clusters by fragmentation (breaking a table into two tables) and coagulation (merging two tables) can be described as a Fragmentation-Coagulation Process (FCP). They show that FCP is exchangeable, reversible, and has asymptotic distribution of CRP.

Priors over recurrent continuous time processes [code]
Ardavan Saeedi, Alexandre Bouchard-Cŏté

This paper received the best student paper award this year, and Ardavan is only a masters student! The problem he is interested in is a discrete latent state dependent continuous time series with partial observation process. For example, a recurrent disease with coarsely quantified states. He introduces the Gamma-exponential process, where an infinite Markovian transition rate matrix prior is given, extends to hierarchical case, and shows how to do inference.

Kernel Beta process
Lu Ren, Yingjian Wang, David Dunson, Lawrence Carin (none of the authors made it to the conference)

Beta process is a distribution over discrete random measures where each “stick” is in , but does not sum to 1 as Dirichlet process (DP) does. In this paper, they smooth the sticks in relation to covariates through a kernel, such that their heights are correlated. Kernel here does not have to be positive definte, but only bounded positive functions (like pdf’s). I’m curious if a similar approach can be taken for DP. This was originally done in similar fashion for DP by Dunson and Park (2008) (‘kernel stick breaking process’).

Sparse estimation with structured dictionaries
David Wipf

Given an ill-posed problem , where the dictionary , and observation is known, under sparsity assumption this can be solved with regularization, when is incoherent (roughly independent columns). However, when the dictionary is more structured, it can cause problems. This paper alleviates this problem by transforming the sparse variables which effectively re-normalizes them. It turns out the solution is similar to iteratively reweighted with a different penality.

Sequence learning with hidden units in spiking neural networks
Johanni Brea, Walter Sen, Jean-Pascal Pfister

Given a point process, the problem is to train a spiking neural network composed of GLM units (including hidden units) that would generate the training patterns. Minimization of KL-divergence between the given point process, and the one parameterized by GLM is done by online gradient descent. The gradient requires marginalization over the spikes of the hidden units: , so they developed an importance sampling scheme where the samples from hidden units are obtained given the training spikes. The resulting training rule is Hebbian, and analogous to STDP. The results are shown when the given distribution is a delta, that is, when the network has to produce exactly one pattern, and that pattern only.

Wednesday

I presented Bayesian Spike Triggered Covariance analysis as a poster:

Thursday

Empirical models of spiking in neural populations
Jakob H. Macke, Lars Büsing, John P. Cunningham, Byron M. Yu, Krishna V. Shenoy, Maneesh Sahani

A comparison study between coupled GLM model and latent variable model (Poisson linear dynamical system) to fit the motor cortex observations (preparation phase only). While GLM explicitly allows only coupled input between the output of the population spiking history, the latent variable model allowed a low dimensional hidden common input source with linear dynamics. They show that the latent variable model fits better and could reconstruct the cross-correlations while the GLM couldn’t. There were quite a bit of discussions on the floor after the oral presentation. The difference in performance was probably due to (1) relatively large bin size (10 ms), (2) neurons were recorded by Utah array which means low probability of direct connectivity. The coupled GLM was successfully applied to retina where the coupling is local, and the sampling of the neurons were very high with 0.1 ms bin size. It would be interesting to see further developments of latent variable models and GLMs in modeling such motor system data.

Workshops

Hierarchical algorithms for χ-armed bandits
Rémi Munos

This was a non-Bayesian invited talk for the Bayesian optimization, experimental design and bandits workshop. He talked about his paper for the main conference “Optimistic Optimization of a Deterministic Function without the Knowledge of its Smoothness”. In this case, smoothness assumption comes from , where is a semi-metric, and hence the function is bounded from below around the global maximum by the semi-metric. Then using a hierarchical partitioning of the input space that respects the semi-metric, one can get a bound of the function. (This assumption is  not weaker nor stronger than Lipschitz continuity, since the absolute value is missing and it is only from the maximum.) When the knowledge of semi-metric is perfect, the convergence rate of the simple regret (best function value) can be exponential (depending on the semi-metric; multiple semi-metrics can give the bound but the convergence rate can differ). When the semi-metric is unknown, and one overestimates the exponent, for example, global convergence is not guaranteed.

Dynamic Batch Bayesian Optimization
Javad Azimi, Ali Jalali, Xiaoli Fern

When parallel experiments are possible, experimental design with batch sampling can improve the efficiency, but sequential design often performs better than batch design. Under the assumption that the maximum of the function has a known bound, and using the GP predictive covariance, they choose a set of points that are loosely independent, and could improve the criterion.

Future information minimization as PAC Bayes regularization in Reinforcement Learning
Naftali Tishby

This was the last invited talk for the New frontiers in model order selection workshop. Tishby talked about reinforcement learning in a POMDP setup, but I couldn’t fully follow (in fact it went over my head mostly). In a perception-action cycle, the Bellman equation describes the world evolution and associated reward, and he describes a counter part for the agent (mental state?) using an associated Bellman equation with information-to-go (mutual information with respect to a goal).  Then he describes reinforcement learning as a coding problem (relating to Kraft’s inequality, which says subtree of an optimal coding tree is an optimal coding tree). At some point, he reaches PAC-Bayesian bound, and claims that reinforcement learning self-regularizes.

Between the philosophy of science and machine learning
David Corfield [U of Kent]

This was the first invited talk for the Philosophy and machine learning workshop. He talked about a broad range of philosophers (of science) and a couple of examples of interaction between ML. The first example was Karl Popper‘s idea of complexity of theory in terms of falsifiable dimensions and its similarity to VC dimension (see their paper in 2009 for details). The second example was Judea Pearl’s use of counterfactual (by David Lewis), and its impact on philosophy of science. He talked about what kinds of sciences can be benefited from ML, certainly the ones with lots of data. He also went through many philosopher’s ideas including: Popper, Carnap, Kuhn and Lakatos, Feyerabend. It is certainly a very fascinating area, but my impression was that we don’t have much to talk about yet.

It essentially estimates the first order Volterra expansion of the neural response function, that is, approximating a neuron as a linear system. Although neuron is not really a linear system, STA works well in practice. Moreover, it is a consistent estimator for a linear-nonlinear Poisson (LNP) model if the stimulus is white Gaussian noise [Bussgang 1952 in Dayan & Abbott]. In [Paninski 2003] this condition is extended to an arbitrary radially symmetric stimulus that induces non-zero mean response.

When the neuron’s features space is in low-dimension, but not 1-dimension, then STA is not sufficient, since it recovers only a 1-dimensional subspace. Spike triggered covariance (STC) is an extension of STA that can consistently estimate filters of a multi-dimensional LNP model [Paninski 2003]. Let us denote the zero-mean stimulus distribution as , and the spike triggered distribution as . Then, STA is the mean of (empirical estimate of ), and STC is the eigen-vectors of the covariance matrix of . STC is only a consistent estimator when the stimulus distribution is Gaussian [for details, see Paninski 2003].

STA/STC are moment based estimators, and does not have a probabilistic model. Analogous to PPCA (probabilistic principal component analysis) provided a generative model for PCA, allowing Bayesian extensions of PCA, we formulate the STA/STC problem as a maximum likelihood estimate of a generative model. Inspired by iSTAC [Pillow & Simoncelli 2006], we extend the LNP model (figure) with exponentiated quadratic nonlinearity. This allows us to put priors on the features, and develop Bayesian estimators. We further extend it to a general family of models, that allows consistent estimation using arbitrary stimulus distribution and flexible class of nonlinearities. This result will be presented at Neural Information Processing Systems (NIPS) 2011. If you are coming to NIPS, it’s poster W88!

• Dayan & Abbott. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. MIT Press 2001
• Paninski. Convergence properties of three spike-triggered analysis techniques. Network: Computation in Neural Systems. v 14 p 437-464. 2003
• Pillow & Simoncelli. Dimensionality reduction in neural models: An information-theoretic generalization of spike-triggered average and covariance analysis. Journal of Vision. v 6 p 414-428. 2006

Hypothesis 1. Neurons are rate coding and the variability is noise

Because of the nature of biological processes, temporal fluctuations of channels, probabilistic behavior or synaptic transmission and so on act as noise sources to the spike train produced by a neuron (e.g. [Carandini]). Therefore, the neuron is a probabilistic unit of processing. The real information can only be retrieved by marginalizing over multiple trials or the population by reducing the noise (or averaging over time). Linear-nonlinear-Poisson (LNP) modeling of neural activity is popular where the instantaneous rate is the signal, and the observation is a result of a Poisson spike generation [Paninski].

Hypothesis 2. Deterministic transformation of random input

On the opposite side of the spectrum, there are theories where each neuron is mostly a deterministic processing unit, and the Poisson like response originates from the variability of the natural statistics of the world combined with the uncontrollable internal state. For example, if a white Gaussian noise is presented to a leaky integrate-and-fire neuron, the resulting firing activity is a renewal process. The resulting inter-spike interval distribution is usually not analytically tractable, but when the firing rate is low (or the inter-spike interval is longer than the membrane time constant), it can be well approximated with a Poisson process [Stevens] (sometimes referred to as deterministic Poisson process).

Hypothesis 3. Maximizing information transfer

If one views sensory system as maximizing the mutual information of the input and output under certain constraints, the spike trains generated by a stochastic input should be close to a Poisson process. This is because Poisson process is the maximum entropy process among homogeneous point processes with the same rate [Rényi]. There are a few STDP like learning rules derived from such principles [Toyoizumi]. To derive these learning theories neurons themselves are not required to be deterministic, rather it is easier to deal with probabilistic models. But in principle, the neurons are trying to make the spike train as homogeneous Poisson like as possible.

Hypothesis 4. Brain uses variability to represent uncertainty of belief

This is the main idea of probabilistic population code (PPC) [Ma]. The source of variability is the uncertainty of the world or the brain’s model (model mismatch) and the population encoding model P(r|s) summarizes it. The posterior belief (represented as Bayesian probability) about the stimulus given the population spike train responses P(s|r) is proportional to P(r|s)P(s) by Bayes rule. If P(r|s) is composed of Poisson neurons with rate proportional to a corresponding population of tuning curves, then the stimulus uncertainty (assuming normal distribution) is represented by the gain of population firing rate. Simple Bayesian inference from multiple sources can be simply done by adding corresponding firing rates. Note that the coding scheme is rate code, but utilizes the population gain and variability to do Bayesian inference.

Hypothesis 5. Sampling hypothesis

Related to PPC, an alternative way of representing a posterior is to sample from it — in this case as a population firing pattern. This is known as the sampling hypothesis [Fiser]. The variability of the neurons is mainly because of the broadness of the posterior. If the brain had no uncertainty about the stimulus, then the posterior will be very narrow, and the firing pattern will have no variability.

These hypotheses are not mutually exclusive, but rather different theories that explain or make use of the “random nature” of spiking.

• Carandini, M. Amplification of trial-to-trial response variability by neurons in visual cortex. PLoS Biology, 2004, 2(9).
• Paninski, L. Maximum likelihood estimation of cascade point-process neural encoding models. Network: Comput. Neural Syst., 2004, 15, 243-262
  
• Stevens, C. & Zador, A. When is an Integrate-and-fire Neuron like a Poisson Neuron? Advances in Neural Information Processing Systems, 1996, 8, 103-109
• Rényi, A. On an extremal property of the poisson process. Annals of the Institute of Statistical Mathematics, 1964, 16, 129-133
• Toyoizumi, T.; Pfister, J.-P.; Aihara, K. & Gerstner, W. Optimality Model of Unsupervised Spike-Timing-Dependent Plasticity: Synaptic Memory and Weight Distribution. Neural Comp., 2007, 19, 639-671
  
• Ma, W. J.; Beck, J. M.; Latham, P. E. & Pouget, A. Bayesian inference with probabilistic population codes. Nature Neuroscience, Nature Publishing Group, 2006, 9, 1432-1438
  
• Fiser, J.; Berkes, P.; Orbán, G. & Lengyel, M. Statistically optimal perception and learning: from behavior to neural representations. Trends in Cognitive Sciences, 2010, 14, 119-130
  

In my biased eyes, there were many posters advocating the diversity/heterogeneity of neuronal populations this year.

II-33. John Cunningham, Mark Churchland, Matthew Kaufman, Krishna V. Shenoy. Extracting rotational structure from motor cortical data
John Cunningham introduced jPCA, a simple yet powerful method that extracts the most rotational linear projection pairs from a time series. I think this may evolve into a primary visualization tool similar to PCA. The method approximates the data as a linear dynamical system where the Frobenius norm (l2 norm of the vectorized matrix) is used to minimize in the space of skew-symmetric matrices . This constraint optimization can be solved easily by a trick.
From dynamical system theory, the real part of eigenvalues of gives rise to convergence or divergence, while the imaginary part of eigenvalues indicate rotation around a fixed point. It was developed to visualize neuronal dynamics that oscillates, but this technique might also be useful for finding limit cycles, or phase synchronized population dynamics in general.

III-32. Mark Churchland, John Cunningham, Matthew Kaufman, Stephen I. Ryu, Krishna V. Shenoy. Firing rate oscillations underlie motor cortex responses during reaching in monkey
This is the sister poster that uses jPCA (II-33) to analyze population dynamics of motor responses. Churchland showed that motor responses of monkey did show rotational dynamics in the later phase, and he believes that the earlier stage of motor planning is to carefully regulate the state of the system at the correct initial condition which is critical for the later dynamics.

T-6. Tony Zador. Cortical circuits underlying auditory processing
He actually changed the title, and outlined an awesome plan to recover complete connectome of the brain. The plan is inspired by brainbow where neurons are colored with randomization of DNA sequences. Instead of visually coloring with fluorescent proteins, his plan is to tag them with unique DNA sequence barcodes, and use viruses to transynaptically jointly tag DNAs by combining pairs of neurons that are connected. Assuming the cost and speed of DNA sequencing technology, we would be able to obtain the full connectome of the brain with high probability pretty soon.

T-14. Tim Vogels, Henning Sprekeler, Friedemann Zenke, Claudia Clopath, Wulfram Gerstner. Inhibitory synaptic plasticity generates global and detailed balance of excitation and inhibition
Inhibitory to excitatory spike timing dependent synaptic plasticity rule that could put a network of leaky integrate and fire neurons in a balanced state was presented. This is without changing the excitatory-to-excitatory synapses. In fact, they showed that you can have local excitation structure embedded that could be recalled in this network.

I-86. Alex Huk, Miriam Meister. Neuronal heterogeneity in LIP: Banburismus in the brain, or enigma machine?
Alex Huk showed a collection of neurons that are responsive to memory task and/or discrimination task. The diversity of the population is huge, in terms of temporal organization. Together they support a prolonged activity that is usually seen in population averaged histograms.

II-40. Shimazaki Hideaki, Emery N. Brown. Constructing a joint time-series model of continuous and Bernoulli/Poisson processes using a copula
Copula is a powerful way of capturing dependencies between variables. There have been a several applications of copula in neural data, but it was always between a same type of data; continuous pairs or discrete pairs. Shimazaki is developing a way of capturing the dependence between continuous varaibles and a point process. So far he can capture instantaneous coupling by using an empirical conditional joint between the continuous variable and the spikes. This is possible because essentially the binary spiking can be considered a binary random variable IMHO.

II-70. Olav Stetter, Demian Battaglia, Jordi Soriano, Theo Geisel. State-dependent network reconstruction from calcium imaging signals
Olav presented their recent effort to enhance measures of causality/connectivity. They are working on analyzing the dynamics of cortical culture. He was not happy with traditional measure such as Granger causality and transfer entropy. He proposed a set of extensions: (1) high-pass filter, (2) capture instantaneous power as well, (3) use transfer entropy, (4) condition on population state. The last extension is very interesting. The network dynamics in this case has clear states:

II-12. Bingni W Brunton, Carlos D Brody. Optimal integration of decision-making evidence in the rat
This is a continuation of the work presented last year at COSYNE by the same authors. Poisson clicks were presented for a different rate on each ear for a fixed duration of 150-800 ms, and the animal (rat) has to decide which side had more clicks. From the psychophysics only, they modeled a drift to decision model with 6 parameters (leakage, sensory noise, accumulator noise, bound, facilitation/depression?). From the likelihood of the parameter space, they conclude that the rat has zero accumulator noise, which means the rats can count pretty well, and keep it in memory pretty well too.

III-33. Wieland Brendel, Naoshige Uchida, Ranulfo Romo, Christian K Machens. How to deal with the heterogeneity of neural responses: A demixing method
Given a set of data that can be conditioned on several categorical conditions, the goal is to find a common set of basis vectors that is related to one of the categorical condition at a time, while explaining the variance. A variant of PCA which is somewhat related to Fisher discriminant is introduced for this purpose they call DPCA. Essentially the conditional covariance matrices (marginalizing out other variables) are computed, and the cost function for PCA is weighted by the fraction of contribution from these covariances to the total.
(update Aug, 2011: Christian Machens has a related paper in Frontiers Computational Neuroscience.)

T-3. Anmo J. Kim , Aurel A. Lazar, Yevgeniy B. Slutski. Drosophila projection neurons encode the acceleration of time-varying odor waveforms
Anmo showed the transfer function of early olfactory processing using precisely controlled temporal odor stimulation. Two serial stages of differentiation effect of the input was demonstrated from olfactory sensory neuron (OSN) and projection neurons (PN). Hence, the effective signal output (spike rate) of PN was the acceleration.

II-76. Alexandra Smolyanskaya, Stephen G Lomber, Richard Born. Individual neurons in MT have significant detect probabilities for motion and depth detection tasks
Using cooling, she could change tuning curve properties of visual cortex. Depth tuning and directional tuning both changed, but with different amounts. Interestingly, the performance of the neurons (measured with detection probability (same as choice probability)) degraded more for depth, if I remember correctly, while the tuning curves were worse for depth.

I-42. Il Memming Park, Miriam Meister, Alexander Huk, Jonathan W Pillow. Detailed encoding and decoding of choice-related information from LIP spike trains
My poster was on the first night (hence less coverage of posters on the same day). We analyzed decision making process from a statistical modeling perspective.

Workshops

Wolf Singer. The role of oscillatory phenomena in sensory processing
He was much more careful to claim the usefulness of temporal code compared to his keynote speech at IJCNN 2007. He gave various examples of dissociation between oscillation/synchrony dynamics and rate codes, but none of them was conclusive evidence that the brain utilizes them to compute. Curiosity, attention, expectancy showed modulation of oscillatory components, and he talked about NIPS 2006 paper on liquid state machine like experiment on visual cortex (Nikolic et al.).

Aysegul Gunduz, Gerwin Schalk. The dynamics of attentional shift reflected in electrocorticography
Aysegul talked about how attention modulates ECoG signal amplitudes in human, and demonstrated that it is possible to decode the spacial attention from ECoG as well. Higher frequency activity on different areas were consistently involved and formed an “attention network”.

On the second day of the workshop, I spent most of my time in the closing the loop: new techniques for online neural characterization and optimal control. The topics were closely related to optimal experiment design, and active learning. Greg Horwitz gave a great opening remark on how the invention of computer gave rise to laziness of the experimentalists. For example, in the old days to find V1 receptive fields, people moved a moving light bulb while listening to the neuron, but now we stimulate with a computer in an open loop most of the time. This workshop was very interesting to me. I hoped that some of the workshops on the second day would have been on the first day though.

Greg Horwitz. Characterizing spatial iso-response surfaces of retinal ganglion cells
In a neural response model where the input is combined into a single quantity, iso-response surface captures how the inputs are combined. In chromatic tuning of V1 neurons, the widely used model is linearly combining each cone output followed by a nonlinearity, which results in a linear manifold in the space of cone contrast. However, when measured adaptively through feedback to find the most non-linear part of the iso-response surface, he found some cells had more quadratic basis, or even elliptical. The adaptive strategy was to model the iso-response surface as a collection of triangles, and refine the details by probing the center of the most non-linear polygon and doing so with interleaving stimulus to avoid adaptation of the neuronal response.

Yashar Amadian. Design of optimal stimuli to control neuronal spiking
Extending last year’s poster of his, he talked about how to use a probabilistic modeling of the system (stimulus to spiking), and constraints (maximum amplitude, smoothness of waveform) to optimize for a stimulation pattern that would produce as precisely timed spike train as the target. The loss function he used for the target was an instantaneous Dirac delta function which would be zero if and only if the target spike occurs within the bin.

Kechen Zhang. Optimal stimulus design and network structure
He talked about neural network theory in system identification of how constraints that bound the parameter space to a closed region in a quadratic neural network model can frequently give rise to at least one parameter on the boundary. This is because the eigenstructure of the quadratic model results in saddles of the response surface, as well as the fact that the continuous map from parameter space to response surface forcing the boundary of the parameter space to be mapped to the boundary in the constrained response surface. He also talked about how the network is ambiguous under certain conditions (power gain function and converging feedforward network).

See also Mark H. Histed and Jonathan W. Pillow’s report about the meeting

I summarized some of the posters that were interesting to me. If there is anything wrong, it’s my fault, and please let me know. You can find all the papers from the online proceedings.

Update: video recordings of the talks [invited talks] [oral sessions].

Day 1

Efficient algorithms for learning kernels from multiple similarity matrices with general convex loss functions
Achintya Kundu, vikram Tankasali, Chiranjib Bhattacharyya, Aharon Ben-Tal

Aiming to learn a positive semi-definite kernel matrix from a given set of similarity measures, and a classification problem, this paper uses subgradient with mirror descent optimization. They combine the SVM cost with loss function on the similarity matrices. For the mirror descent, they chose negative of matrix entropy (unnormalized entropy of eigenvalue distribution) that guarantees  the convergence.

Generative Local Metric Learning for Nearest Neighbor Classification
Yung-Kyun Noh, Byoung-Tak Zhang, Daniel Lee

Using the asymptotic NN-classifier error bound which is measure-independent, the goal is to learn a global Euclidean metric scaling for . The critical problem is that the theoretically measure-independent quantity becomes measure-dependent when estimated from finite samples. By first order expansion of the probability estimated by the nearest neighbor estimator, the authors explicitly derive the bias, and tries to learn the metric while reducing the bias. It is posed as a semi-definite programming (SDP).

Estimation of Rényi Entropy and Mutual Information Based on Generalized Nearest-Neighbor Graphs
David Pal, Barnabas Poczos, Csaba Szepesvari

There have been works on Rényi’s entropy estimation from nearest-neighbor, but this one proves strong convergence results. In addition, they show that given a consistent estimator for Rényi’s entropy, one can use empirical copula to obtain a consistent estimator for Renyi’s mutual information. The fact that the marginals are uniform make it very simple.

Random Conic Pursuit for Semidefinite Programming
Ariel Kleiner, Ali Rahimi, Michael Jordan

According to the authors, interior point method for SDP is slow and tends to jump to the solution instead of slowly converging. As an alternative, they propose a stochastic gradient descent, where for every iteration a random rank 1 positive definite matrix is chosen, and the optimization is solved for the 2-dimensional solution plane instead of the entire cone of semi-definite matrices.

Short-term memory in neuronal networks through dynamical compressed sensing
Surya Ganguli, Haim Sompolinsky

When sparse signal is provided as a single common input to a network with linear recurrent dynamics, the problem of recovering past input signals from current activation pattern can be formulated as an compressed sensing problem. They built a theory of memory depth under this setting where the recurrent dynamics matrix is orthogonal. The assumptions are not very realistic, however, viewing the problem from a compressive sensing point of view connects to the FWIRE project.

Day 2

Machine Learning with HumanIntelligence: Principled Corner Cutting (PC2) – invited talk
Xiao Li Meng

This was an inspiring invited talk with a practical philosophy. Self-consistency principle as a general method to deal with missing data was fresh to me. Compromising statistical principles for more efficient methods that resembles machine learning principles were discussed with personal experiences. [video]

Identifying Dendritic Processing
Aurel A. Lazar, Yevgeniy Slutskiy

Extending the time encoding machine and perfect reconstruction techniques, the authors propose estimating a linear filter followed by a non-noisy integrate-and-fire neuron. Interestingly, due to causality the dendritic process (modeled as a linear filter) has infinite spectrum which conflicts with the (RKHS) theory of band limited functions required for the time encoding machine theory. As a result, they can only recover the filter projected in the band limited space. Moreover, using a clever trick of switching the input and the filter, multiple presentations of different input can be combined to estimate the filter even under the usual Nyquist spiking rate limit.

Near-Optimal Bayesian Active Learning with Noisy Observations
Daniel Golovin, Andreas Krause, Debajyoti Ray

Determining the optimal hypothesis (within a finite set) with a few observation is a problem of active learning. Counter intuitively, the choice that provides maximal information gain do NOT perform well, and in pathological problems the posterior does not eliminate choices. Instead they show the choice that reduces the posterior probability mass the most is adaptive submodular, and works under noisy observations. The main contributions are efficient algorithms that implement such policy.

Robust PCA via Outlier Pursuit
Huan Xu, Constantine Caramanis, Sujay Sanghavi

When PCA is formulated as SVD on the data matrix, outliers are defined as corrupted columns that do not lie in the low dimensional space. Using a combination of nuclear norm (sum of singular values) and L1 norm of the column-wise L2 norm, they propose a convex optimization formulation. The proposed method has deterministic error bound related to the incoherence parameter.

Global Analytic Solution for Variational Bayesian Matrix Factorization
Shinichi Nakajima, Masashi Sugiyama, Ryota Tomioka

The problem of SVD is posed as a generative model. Gaussian model for a low rank matrix is assumed, and solved with analytical solutions using variational Bayes approach of factorizing the posterior of columns of the low rank matrix. Without assistance of symbolic mathematical tools such as mathematica, the authors arrived at analytic solutions with flat prior as well as ARD prior. (The full equations for the solutions were too lengthy to be included in the poster.)

Improvements to the Sequence Memoizer
Jan Gasthaus and Yee Whye Teh

Using hierarchical Pitman-Yor process (a “heavy-tail” version of Dirchlet process with extra parameters) as the prior, the aim is to learn a distribution of a symbol conditioned on all previous symbols. An application is compression where the inference process is replicated in the decoding end with same random seed. I’m new to non-parameteric Bayesian inference, so the details slipped through me, but I would definitely revisit this paper.

Global seismic monitoring as probabilistic inference
Nimar S. Arora, Stuart Russell, Paul Kidwell, Erik Sudderth

From a collection of seismic sensors, they aim to find a probable cause (artificial nuclear activity). Using prior knowledge combined with a heuristic Bayesian inference algorithm, they infer a sequence of events (with position and amplitude) as causes. The inference is done through operations on a Poisson process initialized sequence as a hypothesis and improve it via adding, aligning, reassigning, and pruning; simiar to dependent DP process by Dahua Lin (below). I think a similar process can be used to infer hidden causes of spike train observations in a generative point process modeling.

Day 3

Construction of Dependent Dirichlet Processes based on Poisson Processes
Dahua Lin, Eric Grimson, John Fisher

This paper got the best student paper award. Dirichlet process is related to a compound Poisson process known as Gamma process; Gamma process takes positive values independently distributed as Gamma distribution at each event generated by a Poisson process. Dirichlet process is simply a normalized form of Gamma process, such that the “amplitude” of Gamma process at each point represents the probability of being at that position. It is well known that Poisson process is the only completely random process, hence one can perform operations such as thinning, superposition, and random translation to create a similar Poisson process. The authors create dependent Dirichlet process through these operations and apply it to non-stationary clustering problem by developing Gibbs sampling for it.

A Novel Kernel for Learning a Neuron Model from Spike Train Data
Nicholas K. Fisher, Arunava Banerjee

Neuron modeling is posed as a classification problem for input spike trains that do not elicit an action potential vs. that do. SVM is used with conjunction to a novel spike train kernel of the form . This was derived from a combination of basis functions  for each pair of spike timings they call REEF (reciprocal exponential – exponential function). The kernel is the uniform integration of outer product of REEF. This is a novel binless spike train positive definite kernel that weights the spikes in time non-uniformly (like spikernel). Here they applied to learning SRM₀ (simplest spike response model), but I think this has potential to be applied to arbitrary neuron models.

Universal Kernels on Non-Standard Input Spaces
Andreas Christmann, Ingo Steinwart

Universal kernels have nice convergence properties and learning guarantees (on a compact set); Guassian kernel and Laplacian kernels are widely used because of this property. The authors show that one can construct universal kernels from a simple form . When , then is a universal kernel in a closed ball. Extending this result, if one can find a continuous injective mapping to a separable Hilbert space, by Mercer theorem and the above result, one can build a universal kernel from any compact space. Two interesting examples of kernels on probability space are given; one for characteristic kernel and embedded probability, and the other is using (generalized) characteristic functions of probabilities.

Of course my poster was on the third day as well, so I had little time for other’s posters. (Update: check out http://nipsposterface.com/)

NIPS 2010 starts tomorrow, and we have a poster for the previously mentioned extension of KS and CM test as point process divergences I have developed with Sohan Seth. You can glimpse at the poster below, but as you can see it is fairly compressed. So if you are attending NIPS 2010, please stop by to our poster and we’ll be glad to explain the details (Wednesday Dec 8th, W59, “A novel family of non-parametric cumulative based divergences for point processes”).

One application of a point process divergence is sensory neuralprothesis. The goal of sensory prosthetics is to artificially recreate the activation and inactivation of the brain so that a percept of sensory stimulation would be formed. The current approach is to electrically stimulate the thalamus along the sensory pathway. However, given only a limited number of electrodes, it is difficult to exactly mimic the natural pattern of firing the peripheral nerves. Therefore, a search of stimulation parameter is necessary to find the best possible setting. Austin Brockmeier analyzed a set of stimulation patterns to VPL of anesthetized rat (from Francis Lab) and demonstrated the usefulness and advantage of the proposed divergences compared to mean rate. I am very pleased to see the method working in practice.

The implementation is straightforward, and included in the IOCANE open source project.

Given a product of functions over a probability mass function where K is the cardinality of alphabets, and real valued function , we want to compute the integral of this over the probability simplex ( are non-negative and sum up to 1). The trick is to realize that the integral can be written in convolutions, and use the convolution theorem for Laplace transform.

Note that is not integrated, since it is determined by the rest of the variables. Define . Note that . Rewrite the integral as,

Recognizing the convolution, it can be simplified as,

Using ,

Now, just apply the Laplace transform, and you are good to go!
This trick is useful for Bayesian integrals on Dirichlet prior.

Reference

• David Wolpert, David Wolf. Estimating functions of probability distributions from a finite set of samples. Physical Review E, Vol. 52, No. 6. (December 1995), pp. 6841-6854.

Note that if where is the observation from channel i, is a zero mean random process that is spatially independent, and is the fixed realization of a zero mean random process with variance fluctuation among the channels, the instantaneous cross-correlation is . Hence if are all positive (or all negative), such common additive fluctuation creates positive correlation.

However, if one analyzes the cross-correlation between channels after ensemble mean removal, one would find that there is a tendency that the cross-correlation at zero lag is smaller (often negative) than expected. In essence, this is kind of a small sample size effect. The problem is even when the term is perfectly removed by the ensemble average subtraction, the empirical mean is not zero, and again we are removing this from each channel as well. This can be demonstrated in the simple case where the underlying process are independent and have the same variance, then

.

Therefore ensemble mean introduces negative correlation. The following MATLAB code demonstrates it.

The blue line corresponds to the independent signal, the green the common fluctuation added, and the red the ensemble removed.

N = 1000; % length of time series
M = 20; % number of channels
eta = 0.1 * randn(N,1);
X = randn(N,M);
Y = X + repmat(eta,1,M);
esbm = mean(Y,2); % compute the ensemble mean
Z = Y - repmat(esbm,1,M);
figure; hold all;
[mcc,lags] = meanXcorr(X,10); plot(lags, mcc);
[mcc,lags] = meanXcorr(Y,10); plot(lags, mcc);
[mcc,lags] = meanXcorr(Z,10); plot(lags, mcc);
legend('X', 'Y', 'Z');
where the function meanXcor is simply computes the mean pairwise cross-correlation.
function [mcc, lags] = meanXcorr(X, maxLag)
k = 0;
for n1 = 1:size(X,2)
for n2 = n1+1:size(X,2)
k = k + 1;
[cc(:,k), lags] = xcorr(X(:,n1), X(:,n2), maxLag, 'coeff');
end
end
mcc = mean(cc,2);

To define a measure, we need a measurable space (X, F); a non-empty set X and a σ-algebra F on X. Here X is the space where our stuff to be measured lies, and F gives special structure of the space such that things are well defined and pathological sets can be avoided. An algebra F of  X is a set of subsets of X such that it contains the empty set, and closed under set union and complement. A σ-algebra is an algebra that is closed under countable union. Elements in F are said to be measurable.

A measure μ on (X, F) is a non-negative extended real valued function on F that is countably additive;

where are disjoint measurable sets. Additivity makes sense because we want when A and B are disjoint (number of strawberries should add up for different fields). If μ is always finite valued, then it is called a finite measure. For a special case, a probability measure is a finite measure where μ(X) = 1.

Given a set E, we denote the smallest σ-algebra that contains E as σ(E) and say σ(E) is the σ-algebra generated by E. A measure μ on F is determined by its values on any algebra that generates F (Carathéodory extension theorem).

A predicate P(x) holds almost everywhere μ (or μ-a.e. for short) if it is true except for a set of measure zero, that is, .

A function f from a measurable space (X, F) to a measurable space (Y, G) is measurable if . For a topological space (X, U), the σ-algebra generated by the open sets is called the Borel σ-algebra (or Borel set). Borel set links the measurable functions and continuous functions — every continuous function from X to the real line is measurable with respect to the Borel algebra. When the topological space is induced by a metric, Borel set and Baire set coincide. Baire set is the smallest σ-algebra with respect to which the continuous functions are measurable.

Real-valued Borel-measurable functions are closed under algebraic operations and limit. Moreover, they can be approximated by limit of simple functions. A simple function is a finite linear combination of indicator function of measurable sets. By the linearity of integration, integration of a Borel-measurable function with respect to a measure μ can be defined by letting the integration of an indicator function on the measurable set A as μ(A): .

References:

1. Halmos, P. R. Measure theory. Springer-Verlag, 1974
2. Daley, D. J. & Vere-Jones, D. An Introduction to the Theory of Point Processes. Springer, 1988

P.S. Terry Tao has a great summary on measure theory and integration.

Spike train pairs(A, B) and (B, C) are synchronous as they have synchronized action potential timings (red and blue, respectively). However, A, B, C are not synchronous all together in this example. Hence, there’s a need for capturing this higher-order synchrony structure.

Even when the time window of analysis is small enough, such that in a single trial basis, synchrony in pairs (A,B) and (B,C) implies synchrony in (A,C) because the jitter allowed for synchrony detection is in the order of window size, pairwise synchrony does NOT imply higher-order synchrony because in some trials (A,B) may be synchronous and some other trials (B,C) may be synchronous.

Detection of mutual interaction of multiple neurons is a difficult problem, and we know pairwise analysis tools can only check necessary conditions but not sufficient. (Simple example: given 3 random variables, X, Y, Z, pairwise independence does not imply mutual independence, since P(X,Y,Z) = P(X)P(Y)P(Z) may not hold.)  Nevertheless most analysis are done with pairwise analysis these days. I was impressed by the work by Sonja’s group on CuBIC method to detect the presence of these higher order synchrony [1]. The main idea is that amplitude distribution of the population rate, they call ‘complexity distribution‘, has a heavier tail when the higher order synchrony is present. The idea does not require a combinatorial search, and can be directly applied without spike sorting. The disadvantages are that the assumptions may not hold in real data, and only the presence is detected and the actual interaction is not directly captured.

References

1. Benjamin Staude, Stefan Rotter, Sonja Grün. CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains. Journal of Computational Neuroscience (28 October 2009) dx.doi.org/10.1007/s10827-009-0195-x